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• C(X,Y)
  125 B (14 words) - 05:55, 1 January 2017

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• Homeomorphism
  * Given a bijective continuous map, say {{M|f:X\rightarrow Y}}, the following are equivalent{{rITTMJML}}: ...pological space|topological spaces]] a ''homeomorphism from {{M|X}} to {{M|Y}}'' is a{{rITTMJML}}:
  5 KB (731 words) - 22:58, 22 February 2017
• Bijection
  ...e useful property that for <math>f:X\rightarrow Y</math> that <math>f^{-1}(y)</math> is always defined, and is at most one element. ...e <math>\mathcal{P}(X)</math> denotes the [[Power set|power set]] of <math>X</math>)
  732 B (124 words) - 11:49, 26 September 2016
• Injection
  ...wo distinct things in <math>X</math> are mapped to the same thing in <math>Y</math>. That is<ref name="API">Analysis: Part 1 - Elements - Krzysztof Maur * <math>\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]</math>
  3 KB (463 words) - 21:50, 8 May 2018
• Open ball
  Given a [[metric space]] {{M|(X,d)}} the ''open ball centred at {{M|x_0\in X}} of radius {{M|r>0}}'', denoted {{M|B_r(x_0)}} (however many notations are ...1=B_r(x_0):=\{x\in X\vert\ d(x,x_0)<r\} }} - that is all the points of {{M|X}} that are a distance (given by {{M|d}}) strictly less than {{M|r}} from {{
  4 KB (842 words) - 02:00, 29 November 2015
• Metric space
  A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction * <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes
  2 KB (336 words) - 06:07, 27 November 2015
• Nabla
  ...f{i}\frac{\partial(\ )}{\partial x}+\mathbf{j}\frac{\partial(\ )}{\partial y}+\mathbf{k}\frac{\partial(\ )}{\partial z}</math> ...a^2(\ )=\frac{\partial^2(\ )}{\partial x^2}+\frac{\partial^2(\ )}{\partial y^2}+\frac{\partial^2(\ )}{\partial z^2}</math>
  1 KB (245 words) - 18:35, 13 February 2015
• Continuous map
  ...{J})}} and {{M|(Y,\mathcal{K})}} we say that a [[map]], {{M|f:X\rightarrow Y}} is continuous if<ref name="KMAPI">Krzysztof Maurin - Analysis - Part 1: E * The [[pre-image]] of every set open in {{M|Y}} under {{M|f}} is open in {{M|X}}
  6 KB (972 words) - 01:44, 14 October 2016
• Surjection
  ...ightarrow Z}} be surjective maps, then their composition, {{M|1=g\circ f=h:X\rightarrow Z}} is surjective. : We wish to show that <math>\forall z\in Z\exists x\in X[h(x)=z]</math>
  2 KB (263 words) - 21:56, 8 May 2018
• Dual vector space
  ...hich is to say <math>f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)\ \forall x,y\in V\ \forall \alpha,\beta\in F</math>) ...der (for {{M|v\in\mathbb{R}^2}}: <math>f^*(v)=2x</math> and <math>g^*(v)=y-x</math> - it is easy to see these are linear and thus are covectors!<ref gro
  3 KB (614 words) - 05:35, 8 December 2016
• Connected (topology)
  Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is ''connected'' if{{rITTMJML}}: ...}[U\ne\emptyset\wedge V\neq\emptyset\wedge U\cap V=\emptyset\wedge U\cup V=X])}}
  5 KB (866 words) - 01:52, 1 October 2016
• Compactness
  A [[topological space]], {{M|(X,\mathcal{J})}} is ''compact'' if{{rITTGG}}{{rITTBM}}: * Every [[open covering]] of {{M|X}}, {{M|\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J} }} contains a ''finit
  5 KB (828 words) - 15:59, 1 December 2015
• Subspace topology
  ...let {{M|S}} be a subset of {{M|X}}, possibly empty, possibly equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\ ...|(S,\mathcal{J}_S)}} are precisely the intersection of open sets of {{Top.|X|J}} with {{M|S}}
  6 KB (1,146 words) - 23:04, 25 September 2016
• Quotient topology
  : '''Claim 1: ''' {{M|\mathcal{K} }} is indeed a topology on {{M|\frac{X}{\sim} }} X \ar[r]^p \ar[dr]_f & Q \ar@{.>}[d]^{\tilde{f}}\\
  5 KB (795 words) - 13:34, 16 October 2016
• Motivation for topology
  ...>\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))</math>. It seems natural to ask "what do we really nee <math>\forall\text{open sets}\in Y,\ f^{-1}(\text{that open set})</math> is open. This looks very different fr
  1 KB (243 words) - 15:39, 13 February 2015
• Continuity definitions are equivalent
  ...etric space]] to another is the same as <math>f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})</math> being continuous (where the topologies are those [[Topo ...>\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))</math>
  2 KB (476 words) - 07:20, 27 April 2015
• Closed set
  ...l space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X-A</math> is open<ref>Introduction to topology - Third Edition - Mendelson</ A subset {{M|A}} of the [[Metric space|metric space]] {{M|(X,d)}} is closed if it contains all of its [[Limit point|limit points]]<ref g
  1 KB (238 words) - 15:36, 24 November 2015
• Vector space
  * <math>+:V\times V\rightarrow V</math> given by <math>+(x,y)=x+y</math> - vector addition ...es:F\times V\rightarrow V</math> given by <math>\times(\lambda,x)=\lambda x</math> - scalar multiplication
  2 KB (421 words) - 16:30, 23 August 2015
• Triangle inequality
  ...)</math> (see [[Metric space|metric space]]) of which <math>|x-z|\le|x-y|+|y-z|</math> is a special case. ...et <math>|x-y+y-z|\le|x-y|+|y-z|</math> which is just <math>|x-z|\le|x-y|+|y-z|</math>
  3 KB (546 words) - 13:05, 19 February 2016
• Product topology
  ...on the [[set]] {{M|1=\prod_{\alpha\in I}X_\alpha}} (herein we define {{M|1=X:=\prod_{\alpha\in I}X_\alpha}} for notational convenience, where {{M|\prod_ ...s|a topology to be generated by a basis]], thus yielding a topology on {{M|X}}, and
  5 KB (871 words) - 20:32, 23 September 2016
• Function
  * {{M|f\subseteq X\times Y}} ...\forall x\in X\forall y,z\in Y[(x\mathcal{R}y\wedge x\mathcal{R}z)\implies y=z]}}
  4 KB (659 words) - 13:01, 19 February 2016
• Set theory axioms
  ...|Y}} and every element of {{M|Y}} is also an element of {{M|X}} then {{M|X=Y}}<br/> ...th>\forall X\forall Y(\forall u(u\in X\leftrightarrow u\in Y)\rightarrow X=Y)</math>
  3 KB (619 words) - 10:25, 11 March 2015
• Motivation for set theory axioms
  ...of {{M|X}} then <math>X=Y</math>''' - that is {{M|X}} is identical to {{M|Y}}. This looks a lot like {{M|[X\subset Y\text{ and }Y\subset X]\iff[X=Y]}} - infact this is the axiom we use to to get to this.
  3 KB (584 words) - 23:03, 28 February 2015
• Relation
  * {{M|\mathcal{R}\subseteq X\times Y}} We say that {{M|\mathcal{R} }} is a ''relation in {{M|X}}''<ref name="APIKM"/> if:
  4 KB (762 words) - 20:07, 20 April 2016
• The domain, range and field of a relation exist
  ...e given a relation {{M|R}} and wish to show the set <math>\{x|\exists y:(x,y)\in R\}</math> exists, to do this we require the [[Set theory axioms|axioms
  339 B (63 words) - 07:22, 27 April 2015
• Ordering
  ...h> and <math>y < z</math> (which may be written more compactly as <math>x< y< z</math>) then: #:* <math>x\le y\wedge x\ne y</math>
  5 KB (1,006 words) - 13:21, 1 January 2016
• Equivalence relation
  ...{{M|X}}<ref group="Note">This terminology means {{M|\sim \subseteq X\times X}}, as described on the [[relation]] page.</ref> is an ''equivalence relatio ...|\forall x\in X[(x,x) \in \sim]}}. Which we write {{M|\forall x\in X[x\sim x]}}.
  3 KB (522 words) - 15:18, 12 February 2019
• Bilinear map
  :* If {{M|1=u=0}} then (by definition we have) {{M|1=\forall x\in U[0x=0]}} (note the first 0 is a scalar, the second the 0 vector) :*: Let {{M|x\in U}} be given
  4 KB (682 words) - 15:44, 16 June 2015
• Example comparing bilinear to linear maps
  Take {{M|T:\mathbb{R}\rightarrow\mathbb{R} }} with <math>T(x)=x+x</math><br /> To be a linear map <math>T(ax+by)=aT(x)+bT(y)</math>, so take:
  703 B (131 words) - 16:30, 23 August 2015
• Norm
  ...of {{M|\Vert\cdot\Vert}} could be in {{M|\mathbb{C} }} then the {{M|\Vert x\Vert\ge 0}} would make no sense. What ordering would you use? The [[canonic # <math>\forall x\in V\ \|x\|\ge 0</math>
  6 KB (1,026 words) - 20:33, 9 April 2017
• Cauchy-Schwarz inequality
  * {{MM|1=\vert\langle x,y\rangle\vert\le\Vert x\Vert \Vert y\Vert}} - the rare but more general ...a proof of the second form - note that {{M|\Vert x\Vert:\eq\sqrt{\langle x,x\rangle} }} is the [[norm induced by the inner product]] [[User:Alec|Alec]]
  3 KB (609 words) - 13:04, 4 April 2017
• Euclidean norm
  Here for <math>x\in\mathbb{R}^n</math> we have: <math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math>
  985 B (184 words) - 07:23, 27 April 2015
• Sequential compactness
  A [[Topological space|topological space]] {{M|(X,\mathcal{J})}} is sequentially compact if every (infinite) [[Sequence]] has ...ecalling that a [[Norm|norm]] can give rise to the metric <math>d(x,y)=\|x-y\|</math>
  1 KB (228 words) - 15:37, 24 November 2015
• Basis and coordinates
  ...so the coordinate {{M|(x,y)}} is on our paper, and {{M|(x,y)'}} or {{M|(x',y')}} is on their paper. ...r map|linear transform]]? Well recall to be linear <math>T(ax+by)=aT(x)+bT(y)</math>
  9 KB (1,525 words) - 16:30, 23 August 2015
• Metric subspace
  Given a [[Metric space|metric space]] {{M|(X,d)}} and any {{M|A\subset X}}, we can define a metric as follows: ...>d_A:A\times A\rightarrow\mathbb{R}</math> where <math>d_A(x,y)\mapsto d(x,y)</math> (so a restriction of the function essentially)
  429 B (83 words) - 10:28, 11 May 2016
• Formal logic language
  ...Function|functions]] as such (arguably they are functions, <math>P:X\times Y\rightarrow\{\text{true},\text{false}\}</math>) * <math>\text{Mortal}(\text{Person }x)</math>
  2 KB (410 words) - 16:35, 9 March 2015
• Group
  | <math>\forall g\in G\exists x\in G[xg=gx=e]</math> ...d {{M|(-x)}} for the inverse, {{M|y-x}} is simply a short hand for {{M|y+(-x)}}
  7 KB (1,332 words) - 07:17, 16 October 2016
• Jacobian
  * differential of {{M|f}} at {{M|x}}, denoted <math>df_x</math> or <math>Df_x</math> which I prefer, as you of * Jacobian matrix of {{M|f}} at {{M|x}} often denoted <math>J_{f(x)}</math>
  2 KB (389 words) - 13:45, 12 March 2015
• Extended real value
  ...nction|function]] is described as "extended real valued" it means: <math>f:X\rightarrow \mathbb{R}\cup\{-\infty,+\infty\}</math> ...ng algebraic relations are defined on {{M|-\infty}}, {{M|+\infty}} and {{M|x\in\mathbb{R} }}<ref>
  2 KB (396 words) - 16:07, 13 March 2015
• Additive function
  Here {{M|(X,+_X:X\times X\rightarrow X)}} (which we'll denote {{M|X}} and {{M|+_X}}) denotes a set endowed with a binary operation called addi The same goes for {{M|(Y,+_Y:Y\times Y\rightarrow Y)}}.
  6 KB (971 words) - 18:16, 20 March 2016
• Subgroup
  ...\times H\rightarrow G</math> given by <math>\times_H(x,y)\mapsto\times_G(x,y)</math> has <math>\text{Range}(\times_H)\subseteq H</math> #* That is to say it is closed. <math>\forall x\in H\forall y\in H[\times_H(x,y)\in H]</math>
  2 KB (364 words) - 17:35, 15 March 2015
• Coset
  To say <math>x\in gH</math> is to say <math>\exists y\in H:x=gy</math> that is: *<math>[x\in gH]\iff[\exists y\in H:x=gy]</math>
  3 KB (616 words) - 18:08, 15 March 2015
• Ring
  | <math>\exists e\in R\forall x\in R[e+x=x+e=x]</math> The "exists {{M|e}} forall {{M|x\in R}}" is important, there exists a single {{M|e}} that always works
  7 KB (1,248 words) - 05:02, 16 October 2016
• Motivation for quotient topology
  Quotients (for example {{M|X/\sim}} should already mean "gluing" or "associating" things together (like ...}} so is also {{M|0.5}} up from {{M|(x,0)}} - thus we have {{M|(x,1.5)\sim(x,0.5)}}
  4 KB (681 words) - 10:33, 7 April 2015
• Open and closed maps
  A <math>f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})</math> (which need not be continuous) is said to be '''an open A <math>f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})</math> (which need not be continuous) is said to be '''a close
  4 KB (692 words) - 08:00, 8 April 2015
• Motivation for smooth manifolds
  We will chart <math>\mathbb{R}_{++}^2=\{(x,y)\in\mathbb{R}^2|x>0\wedge y>0\}</math>. ...hinking of this as a part of the plane, and clearly with coordinate {{M|(x,y)}} - coordinates in "the standard basis" to use linear algebra terminology.
  2 KB (267 words) - 18:08, 10 April 2015
• Motivation for tangent space
  ...gs as an angle and a distance from the origin, rather than {{M|x}} and {{M|y}}) We will have two ways of looking at points, as {{M|(x,y)}} - traditionally, and {{M|(r,\theta)}} where:
  6 KB (975 words) - 00:18, 11 April 2015
• Polar equation of a line
  * {{M|1=y=mx+c}} ** this means <math>(x-r_1)(x-r_2) = 0</math> whenever {{M|1=x=r_1}} or {{M|1=x=r_2}}, so:
  1 KB (223 words) - 22:43, 10 April 2015
• Motivation for tangent space definitions
  \frac{\delta r}{\delta x} & \frac{\delta r}{\delta y} \\ \frac{\delta \theta}{\delta x} & \frac{\delta \theta}{\delta y}
  4 KB (790 words) - 22:25, 12 April 2015
• Circle
  ...fined by {{M|1=\mathcal{S}^1=\Big\{(x,y)\in\mathbb{R}^2{{!}}d\Big((0,0),(x,y)\Big)=1 \Big\} }}
  3 KB (592 words) - 16:57, 11 May 2015
• Compact-to-Hausdorff theorem
  ...on between two [[Topological space|topological spaces]] {{M|f:X\rightarrow Y}} where {{M|X}} is [[Compactness|compact]] and {{M|Y}} is [[Hausdorff space|Hausdorff]]
  1 KB (219 words) - 12:36, 13 August 2015
• Relatively open
  ...n'''<ref>Introduction to topology - Third Edition - Mendelson</ref> in {{M|X}} Alternatively we may say given a {{M|A\subseteq X}} the family of sets:
  592 B (97 words) - 18:42, 19 April 2015
• Relatively closed
  ...l space]] {{M|(X,\mathcal{J})}} if {{M|A}} is [[Closed set|closed]] in {{M|Y}}
  406 B (60 words) - 18:40, 19 April 2015
• Inner product
  * <math>\langle x,y\rangle = \overline{\langle y, x\rangle}</math> (where the bar denotes [[Complex conjugate]]) ** Or just <math>\langle x,y\rangle = \langle y,x\rangle</math> if the inner product is into {{M|\mathbb{R} }}
  6 KB (1,016 words) - 12:57, 19 February 2016
• Associative
  ...x(yz)</math> where {{M|xy}} denotes the operator acting on {{M|x}} and {{M|y}} ...ction {{M|\times:S\times S\rightarrow S}} we even call the image of {{M|(x,y)}} under {{M|\times}} the ''product'' (or indeed the ''sum'' if we're using
  455 B (77 words) - 07:44, 27 April 2015
• Monoid
  * [[Associative]] - that is <math>\forall x,y,z\in S[(xy)z=x(yz)]</math> * Has identity element - that is <math>\exists e\in S\forall x\in S[ex=xe=x]</math>
  735 B (131 words) - 07:48, 27 April 2015
• Expected value
  ...riable|random variable]] {{M|X}} we define the '''expected value''' of {{M|X}} as: * <math>\mathbb{E}[X]=\sum x\mathbb{P}[X=x]</math>
  516 B (77 words) - 17:40, 8 May 2015
• Surjection/Definition
  Given a [[Function|function]] {{M|f:X\rightarrow Y}}, we say {{M|f}} is ''surjective'' if: * <math>\forall y\in Y\exists x\in X[f(x)=y]</math>
  273 B (54 words) - 17:42, 10 May 2015
• Kernel
  ...hich is imbued with an identity element), the kernel of {{M|f:X\rightarrow Y}} (where {{M|f}} is a [[Function|function]]) is defined as: ...=\{x\in X|f(x)=e\}</math> where <math>e</math> denotes the identity of {{M|Y}}
  2 KB (376 words) - 19:53, 10 May 2015
• Covector
  ...V\ \forall\alpha,\beta\in F[f^*(\alpha x+\beta y)=\alpha f^*(x)+\beta f^*(y)]</math>
  742 B (124 words) - 10:40, 12 June 2015
• Conjugation
  * {{M|1=\exists x\in G[xgx^{-1}=h]}} Let {{M|x}} in {{M|G}} be given, define:
  3 KB (498 words) - 14:51, 18 May 2015
• Normal subgroup
  * <math>\forall x\in G[xH=Hx]</math> where the {{M|xH}} and {{M|Hx}} are left and right [[Cos ** This is the sameas saying: {{M|1=\forall x\in G[xHx^{-1}=H]}}
  5 KB (1,026 words) - 18:07, 17 May 2015
• Examples of rings
  Let <math>S=\{x+y\sqrt{2}\in\mathbb{R}|x,y\in\mathbb{Z}\}</math>, defining multiplication and addition in the usual wa * As {{M|1=x=(a+b\sqrt{2})\in\mathbb{R} }} and {{M|1=y=(c+d\sqrt{2})\in\mathbb{R} }} we know automatically {{M|xy=yx}} as multipli
  2 KB (269 words) - 17:11, 19 May 2015
• Passing to the quotient (function)
  ...0em;}}Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}}<ref group="Note">I have chosen {{M|W}} to mean "whatever"</r # <math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math>
  8 KB (1,644 words) - 20:49, 11 October 2016
• Notes:ToMond
  You obviously know already that the quotient map, {{M|q:X\rightarrow W}} is the "biggest" map (or makes {{M|W}} the largest topology) X \ar[r]^q \ar[dr]_f & W \ar@{.>}[d]^{\tilde{f}}\\
  5 KB (921 words) - 05:43, 7 June 2015
• Direct sum (ring)
  * {{M|1=R\times S=\{(x,y)\vert\ x\in R\wedge y\in S\} }} * Given {{M|(x,y),\ (x',y')\in R\oplus S}} we define:
  3 KB (549 words) - 14:32, 8 June 2015
• Integral domain
  * A commutative ring, that is: <math>\forall x,y\in D[xy=yx]</math> * <math>\forall x,y\in D[xy=yx]</math>
  2 KB (327 words) - 11:09, 20 February 2016
• Continuity and non-surjective functions
  ...s and loops in a topological space|path]] in a [[topological space]], {{M|(X,\mathcal{J})}}, then a path is simply: * {{M|p:([0,1]\subset\mathbb{R},\vert\cdot\vert)\rightarrow(X,\mathcal{J})}} where {{M|\vert\cdot\vert}} denotes the [[absolute value]]
  3 KB (556 words) - 17:42, 6 September 2015
• Bilinear form
  ...n F[\langle\alpha x+\beta y,z\rangle=\alpha\langle x,z\rangle+\beta\langle y,z\rangle]</math> and ...F[\langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle]</math>
  2 KB (283 words) - 09:48, 9 June 2015
• Linear map/Definition
  ...ll x,y\in U</math> we have <math>T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)</math> * <math>T(x+y)=T(x)+T(y)</math>
  725 B (136 words) - 10:34, 12 June 2015
• Krzysztof Maurin's notation
  | "for all {{M|x}} there follows" | Equiv to {{M|\forall x}}, {{M|x}} may be a statement (eg: {{M|1=x:=y\in Y}})
  1 KB (173 words) - 14:51, 18 June 2015
• Trinary index
  ! class="unsortable" | {{M|X}} ! class="unsortable" | {{M|Y}}
  964 B (165 words) - 20:55, 22 June 2015
• Topological property theorems
  ** Notice this is given X is compact, then Y is compact ** Notice this is given X is compact, Y is Hausdorff, f bijective THEN homeomorphism
  3 KB (616 words) - 08:37, 1 July 2015
• Isometry
  ...|lin map]] {{M|L:U\rightarrow V}}<br/> we have {{M|1=\Vert Lx\Vert_V=\Vert x\Vert_U}} ...phism]] {{M|f:(X,d)\rightarrow(Y,d')}}<br/> we have {{M|1=d(x,y)=d'(f(x),f(y))}}<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici<
  1 KB (212 words) - 13:13, 9 July 2015
• Inner product space
  ...{M|\mathbb{R} }} or {{M|\mathbb{C} }}, which we shall denote {{M|F}}) {{M|(X,F)}}, equipped with an We denote this {{M|(X,\langle\cdot,\cdot\rangle,F)}} or just {{M|(X,\langle\cdot,\cdot\rangle)}} if the field is implicit.
  949 B (161 words) - 21:08, 11 July 2015
• Euclidean n-space
  #* Defined by {{M|1=X=\mathbb{C}^n}} (so {{M|X}} consists of all {{n|[[Tuple|tuples]]}} of the form {{M|(x_1,\cdots,x_n)}} #* Defined by {{M|1=X=\mathbb{R}^n}} (so {{M|X}} consists of all {{n|[[Tuple|tuples]]}} of the form {{M|(x_1,\cdots,x_n)}}
  2 KB (398 words) - 14:17, 12 July 2015
• Inequalities for inner products
  ...\vert\le\Vert x\Vert\Vert y\Vert}} for {{M|1=\Vert x\Vert:=\sqrt{\langle x,x\rangle} }} (equality if ''lin dependent'') ...]] we have {{MM|1=\Vert x+y\Vert^2+\Vert x-y\Vert^2=2\Vert x\Vert^2+2\Vert y\Vert^2}}
  1 KB (214 words) - 14:52, 12 July 2015
• Perpendicular
  ...ace|i.p.s]], {{M|(X,\langle\cdot,\cdot\rangle)}} we define {{M|x}} and {{M|y}} being perpendicular<ref name="FA">Functional Analysis - George Bachman an * {{M|1=\langle x,y\rangle=0}}
  346 B (54 words) - 14:56, 12 July 2015
• Index of spaces
  ...aces, or Banach spaces... ) we use the key {{C|obj}} for these. So {{M|L(X,Y)}} becomes {{C|L ( obj obj )}} | {{M|L(X,Y)}}
  3 KB (612 words) - 21:06, 29 February 2016
• Space of square-summable sequences
  * For {{M|x,y\in l_2}} we define {{M|1=\langle x,y\rangle:=\sum^\infty_{n=1}x_i\overline{y_i} }}
  893 B (141 words) - 15:47, 12 July 2015
• Orthonormal set
  Given an [[Orthogonal set|orthogonal set]], {{M|S\subset X}}, where {{M|X}} is an [[Inner product space|i.p.s]], we say {{M|S}} is ''orthonormal''<re * {{M|\forall x\in S}} we have {{M|1=\Vert x\Vert=1}}
  1 KB (154 words) - 16:12, 12 July 2015
• Trace sigma-algebra
  ...ny [[subset]] of {{M|X}}, then we may construct a {{sigma|algebra}} on {{M|Y}} called the ''trace {{sigma|algebra}}'', {{M|\mathcal{A}_Y}} given by{{rMI * {{M|1=\mathcal{A}_Y:=\left\{Y\cap A\ \vert A\in\mathcal{A}\right\} }}
  921 B (144 words) - 12:00, 23 August 2018
• Discrete metric and topology
  ...n ball|open balls]] of {{M|X}} with the discrete topology are entirely {{M|X}} or a single point, that is: ...\in X\vert\ d(x,p)<r\}=\left\{\begin{array}{lr}\{x\} & \text{for }r\le 1\\ X & \text{otherwise}\end{array}\right.}}
  3 KB (482 words) - 18:10, 25 April 2017
• Discrete metric and topology/Metric space definition
  Let {{M|X}} be a set. The ''discrete''{{rITTGG}} metric, or ''trivial metric''<ref>Fu ...ge 0} }} with {{MM|1=d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right. }}
  1,004 B (160 words) - 06:08, 27 November 2015
• Every map from a space with the discrete topology is continuous
  Given two [[Topological space|topologies]], {{M|(X,\mathcal{P}(X))}} and {{M|(Y,\mathcal{J})}} where: ...\mathcal{P}(X))}} denotes the [[Discrete topology|discrete topology on {{M|X}}]]
  3 KB (534 words) - 13:07, 19 February 2016
• Min/max
  * {{MM|1=\max(x,y):=\left\{\begin{array}{lr}x & \text{if }x\ge y\\ y & \text{otherwise}\end{array}\right.}} * {{MM|1=\min(x,y):=\left\{\begin{array}{lr}x & \text{if }x\le y\\ y & \text{otherwise}\end{array}\right.}}
  884 B (144 words) - 02:51, 3 August 2015
• Notes:Lambda calculus
  ...{{M|M}} is any {{lambda|term}} and {{M|x}} any variable then {{M|(\lambda x.M)}} is a {{lambda|term}} * {{M|(\lambda v_0.(v_0v_{00}))}} - a function that {{M|x\mapsto x(v_{00})}} I think
  4 KB (832 words) - 21:22, 11 August 2015
• Compactness/Uniting covers proof
  ...logical space]] {{M|(X,\mathcal{J})}} is [[Compactness|compact]] (when {{M|Y}} is imbued with the [[Subspace topology|subspace topology]]) * Every [[Covering|cover]] by sets open in {{M|X}} has a finite subcover. }}
  7 KB (1,411 words) - 19:44, 15 August 2015
• Notes:Pick function
  ...4}} (projection AND view operator) you are using (that is for a point {{M|x}}, {{M|Px}} is the complete transformation to clip coordinates) Given a point in the world, {{M|x}} this matrix applies the camera position/angle transformation, then the pr
  4 KB (686 words) - 01:43, 15 September 2015
• Notes:Spherical coordinates
  ...cap}} denotes intersection of sets, {{M|x\in A\cap B\iff x\in A\text{ and }x\in B}}, {{M|\emptyset}} denotes the empty set, so here we are saying "there Then for an {{M|x\in U\cap V}} we have two sets of "coordinates", we have:
  10 KB (1,899 words) - 18:48, 23 September 2015
• Category
  ...}<ref group="Note">Munkres calls the class of objects {{M|X}} and uses {{M|X}} for specific objects. Not sure why, so checked definition with [[https:// # For every ordered pair, {{M|(X,Y)}} of ''objects'' a set {{M|\hom(X,Y)}} of ''morphisms'' {{M|f}}
  2 KB (347 words) - 00:36, 27 September 2015
• Notation
  * ''Conventions'' introduce common conventions (shockingly), for example {{M|(X,\mathcal{J})}} for a topological space and other common things. * {{M|f:X\rightarrow Y}}
  729 B (99 words) - 23:41, 8 October 2015
• Function (notation)
  ...sociates with each {{M|x\in X}} a {{M|y\in y}}. We write this as {{M|1=y=f(x)}} ...w Y}} where {{M|X}} is some sort of space (with structure {{M|A}}) and {{M|Y}} is some sort of space with a structure {{M|B}}.
  2 KB (416 words) - 00:05, 9 October 2015
• Sandbox
  A \ar[r]^-f & B\otimes B\otimes B \ar[r]^g & X \\ A \ar[r]^f & B\otimes B\otimes B \ar[r]^g & X \\
  695 B (132 words) - 22:15, 26 October 2015
• Notes:Quotient
  ...t and let {{M|\sim}} be an [[equivalence relation]] on the elements of {{M|X}}. * Then {{M|\frac{X}{\sim} }} denotes the "[[equivalence class|equivalence classes]]" of {{M|~}
  3 KB (588 words) - 09:38, 24 November 2015
• Quotient vector space
  ...|1=[u]+[v]=\pi(\pi^{-1}([u])+\pi^{-1}([v]))=\underbrace{[x\in\pi^{-1}([u])+y\in\pi^{-1}([v])]}_\text{Well-defined-ness}=[u+v]}}<ref group="Note">This is ...{{M|1=\alpha[v]=\pi(\alpha\pi^{-1}([v]))=\underbrace{[\alpha x\ \text{for }x\in\pi^{-1}([v])]}_\text{well-defined-ness}=[\alpha v]}}<ref group="Note">No
  5 KB (879 words) - 23:09, 1 December 2016
• Cauchy sequence/Definition
  ...hich way we consider them (as {{M|n>m}} or {{M|m>n}}) for {{M|1=d(x,y)=d(y,x)}} - I use the ordering to give the impression that as {{M|n}} goes out ahe
  1,000 B (180 words) - 13:55, 5 December 2015
• Limit (sequence)/Discussion of definition
  The idea is that defining "tends towards {{M|x}}" is rather difficult, to sidestep this we just say "we can get as close a ...N}}) where '''''all''''' points after are ''within'' {{M|\epsilon}} of {{M|x}} (where {{M|d(\cdot,\cdot)}} is our notion of distance)
  2 KB (442 words) - 13:38, 5 December 2015
• Cartesian product
  Given two sets, {{M|X}} and {{M|Y}} their ''Cartesian product'' is the set: * {{M|1=X\times Y:=\{(x,y)\ \vert\ x\in X\wedge y\in Y\} }}, note that {{M|(x,y)}} is an ''[[ordered pair]]'' traditionally this means
  2 KB (318 words) - 14:25, 19 February 2016
• Partial ordering
  ...name of the relation, so {{M|(x,y)\in \sqsubseteq}} means {{M|x\sqsubseteq y}} - as usual for [[relation|relations]]</ref>) we say {{M|\sqsubseteq}} is ...,x)\in\sqsubseteq]}} or equivalently<br/>{{M|1=\forall x\in X[x\sqsubseteq x]}}
  4 KB (740 words) - 10:11, 20 February 2016
• Strict partial ordering
  ...the name of the relation, so {{M|(x,y)\in \sqsubset}} means {{M|x\sqsubset y}} - as usual for [[relation|relations]]</ref>) we say that {{M|\sqsubset}} | {{M|\forall x\in X[(x,x)\notin\sqsubset]}} or equivalently:
  3 KB (436 words) - 10:15, 20 February 2016
• Overview of partial orders
  ...n]] on a set {{M|X}}, which we shall call {{M|\mathcal{R}\subseteq X\times X}} that is{{rAPIKM}}{{rRAAAHS}}{{rSTTJ}}: ...,x)\in\mathcal{R}]}} or equivalently<br/>{{M|1=\forall x\in X[x\mathcal{R} x]}}
  3 KB (454 words) - 07:40, 11 April 2016
• Norm/Heading
  * {{M|1=d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert}} * {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle} }}
  1 KB (194 words) - 19:28, 25 January 2016
• Strong derivative/Infobox
  ...M|U}} [[open set|open]] in {{M|X}}<br/><br/>{{M|df\vert_{x_0}:X\rightarrow Y}} a [[linear map]] called the<br/>"''derivative of {{M|f}} at {{M|x_0}}''"
  626 B (112 words) - 14:06, 13 November 2016
• Strong derivative
  ...[normed space|normed vector spaces]], {{M|(X,\Vert\cdot\Vert_X)}} and {{M|(Y,\Vert\cdot\Vert_Y)}} ...[mapping]], {{M|f:U\rightarrow Y}} where {{M|U}} is an [[open set]] of {{M|X}}
  1 KB (212 words) - 11:55, 11 March 2016
• Inner product/Infobox
  * {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle} }} * {{M|1=d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle} }}<br/>
  1 KB (182 words) - 13:25, 14 February 2016
• Metric/Infobox
  ...2em;">{{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }}</span><br/>Where {{M|X}} is any [[set]] * {{M|1=d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert}}
  1 KB (180 words) - 10:39, 11 March 2016
• TestNavbox
  |group2=X |list2=Y
  1 KB (132 words) - 20:11, 25 January 2016
• Covariant functor
  * 3 objects, {{M|X}}, {{M|Y}} and {{M|Z}} in a [[category]] {{M|\mathscr{C} }} ...sms {{M|f:X\rightarrow Y}}, {{M|g:Y\rightarrow Z}} and the morphism {{M|gf:X\rightarrow Z}} corresponding to the [[composition]] {{M|g\circ f}}
  1 KB (205 words) - 16:27, 2 February 2016
• Covariant functor/Definition
  ...\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & TX \end{array}\right. }} Thus if {{M|f:X\rightarrow Y}} and {{M|g:Y\rightarrow Z}} are morphisms of {{M|C}}, then the following [[commutative d
  2 KB (253 words) - 15:47, 2 February 2016
• Contravariant functor/Definition
  ...\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & SX \end{array}\right. }} Thus if {{M|f:X\rightarrow Y}} and {{M|g:Y\rightarrow Z}} are morphisms of {{M|C}}, then the following [[commutative d
  2 KB (263 words) - 16:53, 2 February 2016
• Preorder
  ...set {{M|X}} is a [[relation]] in {{M|X}}, so {{M|\preceq\subseteq X\times X}}, that is both{{rAITCTHS2010}}: ...\in X[(x,x)\in\preceq]}} or equivalently<br/>{{M|1=\forall x\in X[x\preceq x]}}
  2 KB (355 words) - 10:13, 20 February 2016
• Bounded
  ...}} and {{M|(Y,\Vert\cdot\Vert_Y)}} and a [[linear map]] {{M|L:X\rightarrow Y}} we say that ''{{M|L}} is bounded'' if: ** {{M|\exists A>0\ \forall x\in X[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X]}}
  506 B (80 words) - 19:44, 27 May 2016
• Bounded linear map
  ...}} and {{M|(Y,\Vert\cdot\Vert_Y)}} and a [[linear map]] {{M|L:X\rightarrow Y}}, we say that{{rAPIKM}}: ** {{M|\exists A\ge 0\ \forall x\in X\left[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X\right]}}
  618 B (104 words) - 21:30, 19 April 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere
  ...}} and {{M|(Y,\Vert\cdot\Vert_Y)}} and a [[linear map]] {{M|L:X\rightarrow Y}} between them, then the following are equivalent (meaning if you have 1 yo # If we have a sequence {{M|1=(x_n)_{n=1}^\infty\subseteq X}} with {{M|x_n\rightarrow 0}} then {{M|1=(\Vert L(x_n)\Vert_Y)_{n=1}^\infty
  3 KB (491 words) - 01:35, 28 February 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/1 implies 2
  ...d {{M|(Y,\Vert\cdot\Vert_Y)}} and also a [[linear map]] {{M|L:X\rightarrow Y}} then we have: * {{M|L}} is continuous at some {{M|p\in X}}
  5 KB (1,064 words) - 02:24, 28 February 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/2 implies 3
  ...d {{M|(Y,\Vert\cdot\Vert_Y)}} and also a [[linear map]] {{M|L:X\rightarrow Y}} then we have: * If {{M|L}} is continuous at a point (say {{M|p\in X}}) '''then'''
  6 KB (1,091 words) - 00:37, 28 February 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/3 implies 4
  ...d {{M|(Y,\Vert\cdot\Vert_Y)}} and also a [[linear map]] {{M|L:X\rightarrow Y}} then we have: ** {{M|\exists A\ge 0\ \forall x\in X[\Vert L(x)\Vert_Y \le A\Vert x\Vert_X]}} then:
  3 KB (611 words) - 01:17, 28 February 2016
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/4 implies 1
  ...d {{M|(Y,\Vert\cdot\Vert_Y)}} and also a [[linear map]] {{M|L:X\rightarrow Y}} then we have: * Let {{M|1=(x_n)_{n=1}^\infty\rightarrow x}} be a [[sequence]] that [[convergence (sequence)|converges]] - we must sho
  2 KB (279 words) - 01:34, 28 February 2016
• Derivative (analysis)
  Note to self: don't forget to mention the {{M|h}} or {{M|x-x_0}} thing doesn't matter Let {{M|U}} be an [[open set]] of a [[Banach space]] {{M|X}}, let {{M|Y}} be another Banach space.
  3 KB (497 words) - 10:46, 11 March 2016
• Reverse triangle inequality
  Given a [[normed space]] {{M|(X,\Vert\cdot\Vert)}} other than the defining properties of a [[norm]] we also * If {{M|\vert x\vert \le y}} then we have:
  2 KB (306 words) - 09:28, 13 March 2016
• Continuous map/Claim: continuous iff continuous at every point
  ...wo [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} is ''continuous'' {{iff}} it is ''continuous at every point' ...thcal{K}[f^{-1}(\mathcal{O})\in\mathcal{J}]\right)\iff\left(\forall x_0\in X\forall N\text{ neighbourhood to }f(x_0)[f^{-1}(N)\text{ is a neighbourhood
  726 B (109 words) - 16:10, 23 March 2016
• Notes:Continuous at a point
  ...{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} is continuous at {{M|x_0\in X}} if: * {{M|\forall N\subseteq Y\text{ neighbourhood to }f(x_0)[f^{-1}(N)\text{ is a neighbourhood of }x_0]}
  1 KB (238 words) - 20:15, 23 March 2016
• Algebra (linear algebra)
  :# {{M|1=P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)}} and ...ha P(x,y)+\beta P(x,z)}} for all {{M|\alpha,\beta\in F}} and for all {{M|x,y\in V}}</ref> called the "product".
  2 KB (351 words) - 05:26, 1 January 2017
• Monotonic
  ...ightarrow Y}} between two [[poset|posets]], {{M|(X,\sqsubseteq)}} and {{M|(Y,\preceq)}} is ''monotonic'' or ''monotone'' if: * {{M|1=\forall a,b\in X[a\sqsubseteq b\implies f(a)\preceq f(b)]}}, or in words:
  1 KB (190 words) - 04:50, 9 April 2016
• Extension
  An ''extension'' of a [[mapping]], {{M|f:X\rightarrow Y}} is a new function, say {{M|\bar{f}:A\rightarrow B}} where: * {{M|X\subseteq A}} and {{M|Y\subseteq B}} such that:
  575 B (89 words) - 20:02, 8 April 2016
• Restriction
  ...hcal{P}(X)}} denotes the [[power-set]] of {{M|X}}</ref> (so {{M|A\subseteq X}} - and is any subset) we define a new [[function]], the ''restriction of { * {{M|1=f\vert_A:A\rightarrow Y}} by {{M|f\vert_A:a\mapsto f(a)}}
  652 B (107 words) - 20:01, 8 April 2016
• Extending pre-measures to outer-measures
  ...ing for that set]] we see {{M|\forall x\in\beta_A\exists y\in\gamma_A[y\le x]}} - we may now [[passing to the infimum|pass to the infimum]].
  11 KB (1,921 words) - 16:59, 17 August 2016
• Greater than or equal to
  ...silon form|Epsilon form]]:''' {{M|1=x\ge y\iff\forall\epsilon>0[x+\epsilon>y]}}
  1 KB (152 words) - 15:56, 9 April 2016
• Greater than or equal to/Epsilon form
  * {{M|1=x\ge y\iff\forall\epsilon>0[x+\epsilon>y]}} {{M|1=x\ge y\implies\forall\epsilon>0[x+\epsilon>y]}}
  1 KB (249 words) - 15:07, 9 April 2016
• Notes:Quotient topology
  ...Addendum: ''' for some reason I lie here, the author considers a map {{M|f:X\rightarrow A}} where {{M|A}} is a [[set]] and later applies the equivalence ...map, {{M|\pi:X\rightarrow\frac{X}{\sim} }} that takes {{M|\pi:x\rightarrow[x]}}
  6 KB (1,087 words) - 19:45, 26 April 2016
• Topological separation axioms
  However, suppose that a [[topological space]], {{Top.|X|J}} is [[Hausdorff space|Hausdorff]] say, now we have a "strong" property ( (Here {{Top.|X|J}} is a [[topological space]])
  4 KB (569 words) - 00:08, 4 May 2016
• Notes:IPv6
  ...ks / 4 bytes of data may be specified in {{C|w.x.y.z}} form, where {{C|w,x,y}} and {{M|z}} are numbers in the range 0 to 255 inclusive. This is an IPv4 ...atter what {{C|X}} is (as long as the upper two bits are {{C|10}}) and {{C|Y}} can take any value.</ref>
  5 KB (837 words) - 06:12, 24 April 2016
• Quotient topology/Mapping to a set definition
  ...', {{M|\mathcal{K}\subseteq\mathcal{P}(Y)}} is a topology we define on {{M|Y}} as follows: * {{M|\forall U\in\mathcal{P}(Y)[Y\in\mathcal{K}\iff h^{-1}(U)\in\mathcal{J}]}} or equivalently:
  839 B (138 words) - 14:43, 25 April 2016
• Notes:Universal property of the quotient topology
  ...ext-align:center;" | <m>\xymatrix{ X \ar[d]_q \ar@{->}[dr]^{f\circ q} & \\ Y \ar[r]_f & Z}</m> * Suppose {{Top.|X|J}} and {{Top.|Y|K}} are topological spaces
  2 KB (295 words) - 15:44, 25 April 2016
• Passing to the quotient (topology)/Statement
  ...x{ X \ar[d]_\pi \ar[dr]^f & \\ \frac{X}{\sim} \ar@{.>}[r]^{\overline{f} }& Y}</m></span></center> ...}|Q}} be the resulting [[quotient topology]] and {{M|\pi:X\rightarrow\frac{X}{\sim} }} the resulting [[quotient map (topology)|quotient map]], then:
  2 KB (277 words) - 20:23, 11 October 2016
• Notes:Homotopy terminology
  Homotopy is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subseteq\mathbb{R} Here {{M|X}} and {{M|Y}} are [[topological space|topological spaces]]
  2 KB (289 words) - 09:00, 31 October 2016
• Homotopy
  ...et containing all subsets of {{M|X}}; {{M|A\subseteq X\iff A\in\mathcal{P}(X)}}.</ref> a ''homotopy (relative to {{M|A}})'' is any [[continuous function * {{M|H:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }}) such that:
  3 KB (512 words) - 19:31, 26 November 2017
• The stages of a homotopy are continuous
  ...times I\rightarrow Y}} (for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}, where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mat ...{{M|t\in I}} the map {{M|f_t:X\rightarrow Y}} given by {{M|f_t:x\mapsto F(x,t)}} is [[continuous]].
  974 B (168 words) - 23:46, 2 May 2016
• Adjunction
  ...spaces together along a function, denoted {{M|1=X\cup_f Y:=\frac{X\coprod Y}{\langle a\sim f(a)\rangle} }}
  206 B (30 words) - 00:13, 7 August 2016
• Adjunction topology
  ...d topological subspace|closed subspace]] of {{M|Y}} and {{M|f:A\rightarrow X}} is a [[continuous map]], then: ...M|Y}} to {{M|X}} along {{M|f}}''<ref name="ITTMJML"/>, denoted {{M|X\cup_f Y}} is given by<ref name="ITTMJML"/>:
  1 KB (209 words) - 00:12, 7 August 2016
• Lebesgue number
  ...} be a [[metric space]], and {{M|\mathcal{U} }} be a [[open cover]] of {{M|X}}. We define the ''Lebesgue number''{{rITTMJML}} as follows: ...hbb{R} }} such that {{M|\delta>0}} such that {{M|1=\forall A\in\mathcal{P}(X)\ \exists U\in\mathcal{U}[\text{Diam}(A)<\delta\implies A\subseteq U]}}, th
  1 KB (214 words) - 07:44, 10 May 2016
• A continuous map induces a homomorphism between fundamental groups
  ...]] between the [[fundamental group|fundamental groups]] of {{M|X}} and {{M|Y}}{{rITTMJML}}. * We denote this induced homomorphism, {{M|f_*:\pi_1(X,p)\rightarrow\pi_1(Y,f(p))}} and it is given by {{M|f_*:[g]\mapsto[f\circ g]}}
  877 B (135 words) - 03:55, 14 December 2016
• The relation of path-homotopy is preserved under composition with continuous maps
  ...then both {{M|f\circ g_0}} and {{M|f\circ g_1}} are path-homotopic in {{M|Y}}{{rITTMJML}}.
  753 B (122 words) - 12:49, 10 May 2016
• Euclidean metric
  * {{MM|1=d(x,y):=\sqrt{\sum_{i=1}^n(x_i-y_i)^2} }}
  435 B (68 words) - 11:39, 11 May 2016
• LibSR:Proof of concept
  ...ad {{C|1=z=x*y}} not {{C|varyings.assign("z",MAT4F::Multiply(varyings.get("y"),varyings.get("z"));}} or something, and statically typed.
  1 KB (215 words) - 08:18, 1 October 2017
• Homotopic maps
  ...[map|maps]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. ...(relative to {{M|A}}) is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[unit interval]]) such t
  4 KB (674 words) - 13:26, 15 September 2016
• Notes:Infimum
  ...A[y\preceq a]\} }_\text{the set of all lower bounds}[\text{inf}(A)\succeq x]}} ...1=\forall x\in X[(\forall a\in A[x\preceq a])\implies \text{inf}(A)\succeq x]}}
  683 B (131 words) - 23:01, 23 May 2016
• Notes:First order language
  ...then both {{M|\forall xA}} and {{M|\exists xA}} are formulas. We call {{M|x}} a ''bound variable'' : {{Note|Crap is now possible; like {{M|(x\doteq y)\wedge(x\not\doteq y)}}, however there are 2 kinds of crap, the first is that which is ''always'
  6 KB (1,088 words) - 09:22, 28 August 2016
• TrialEq/Documentation
  ...iint_{\sigma}f(x,y,z)\mathrm{d}A}{\sqrt{\displaystyle\sum_{i=1}^n(x_i-\bar{x})^2 } } }}<span style="display:block;float:left;vertical-align:middle;">{{T ...iint_{\sigma}f(x,y,z)\mathrm{d}A}{\sqrt{\displaystyle\sum_{i=1}^n(x_i-\bar{x})^2 } } }}<span style="position:absolute;width:40em;">{{TrialEq|def=1.4}} a
  1 KB (181 words) - 06:05, 26 May 2016
• Notes:Mean value theorem
  ====For {{M|f:[x,x+h]\subseteq U\rightarrow\mathbb{R} }} for {{M|U}} open in a [[Banach space] ...M|x}} to {{M|x+h}} as {{M|a}} goes from {{m|0}} to {{M|1}}. A line in {{M|(X,\Vert\cdot\Vert)}}.
  3 KB (529 words) - 08:07, 4 June 2016
• Notes:Connected space
  * A [[topological space]], {{Top.|X|J}}, is said to be ''disconnected'' if it can be expressed as the [[union]] ** Any such subsets are said to ''disconnect'' {{M|X}}<ref name="ITTMJML"/>.
  1 KB (188 words) - 21:33, 30 September 2016
• Homotopy (object)
  ...et containing all subsets of {{M|X}}; {{M|A\subseteq X\iff A\in\mathcal{P}(X)}}.</ref> a ''homotopy (relative to {{M|A}})'' is any [[continuous function * {{M|H:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }}) such that:
  2 KB (296 words) - 04:25, 1 July 2016
• Notes:Group isomorphism theorems
  ...require {{M|1=\forall x,y\in G[(\pi(x)=\pi(y))\implies(\varphi(x)=\varphi(y))]}} which is not immediately obvious.
  4 KB (654 words) - 23:06, 10 July 2016
• Notes:Equivalence relations
  * {{M|1=x\sim y\implies \pi(x)=\pi(y)}} Let {{M|X}} be a set and let {{M|\sim}} be an [[equivalence relation]], then:
  1 KB (220 words) - 16:56, 12 July 2016
• Group factorisation theorem
  ...and by hypothesis we have {{M|1=[\pi(x)=\pi(y)]\implies[\varphi(x)=\varphi(y)]}} | Let {{M|f:X\rightarrow Y}} and {{M|w:X\rightarrow W}} be [[function|functions]].
  7 KB (1,195 words) - 22:55, 3 December 2016
• Notes:Proof of the first group isomorphism theorem
  ...ppose {{M|1=\theta(x)=\theta(y)}}, we wish to show that this means {{M|1=x=y}}
  3 KB (528 words) - 17:41, 16 July 2016
• Free monoid generated by
  Given a [[set]], {{M|X}}, there is a ''free'' [[monoid]], {{M|(F,*)}}{{rAAPAG}}. ...all the finite [[tuple|tuples]], {{M|(x_1,\ldots,x_n)}} (where {{M|x_i\in X}})
  2 KB (419 words) - 16:20, 20 July 2016
• Semigroup
  ...ive (binary operation)|associative]] - {{M|1=\forall x,y,z\in S[(x*y)*z=x*(y*z)]}}
  631 B (99 words) - 07:27, 21 July 2016
• Notes:Generalising the limit
  ...forward property''''', {{M|(P,\preceq)}} and a [[map]], {{M|f:P\rightarrow X}} where: * {{M|f:P\rightarrow X}} is just a map, there are no extra conditions
  6 KB (1,118 words) - 11:34, 30 July 2016
• Notes:The foundations of Mathematics - Kunen
  | {{M|1=\exists x(x=x)}} | {{M|1=\forall z(x\in x\leftrightarrow z\in y)\rightarrow x=y}}
  2 KB (342 words) - 02:38, 31 July 2016
• Task:Continuity types
  # {{M|f:\mathbb{R}\rightarrow\mathbb{R} }} is continuous at {{M|x\in\mathbb{R} }} if: ...\delta>0\forall a\in\mathbb{R}[\vert a-x\vert < \delta\implies\vert f(a)-f(x)\vert < \epsilon]}} - and a discussion of why this is intuitive, and how to
  3 KB (668 words) - 22:38, 4 August 2016
• Task:Characteristic property of the subspace topology
  ...>\xymatrix{ Y \ar[r]^f \ar[dr]_{i_S\circ f} & S \ar@{_{(}->}[d]^{i_S} \\ & X}</m></center> ...p.|S|K}} say. Suppose that {{Top.|Y|L}} is any topological space and {{M|f:Y\rightarrow S}} is a [[map]]. Then{{rITTMJML}}:
  4 KB (726 words) - 02:59, 7 August 2016
• Topological group
  # {{M|m:G\times G\rightarrow G}} with {{M|m:(x,y)\mapsto x*y}} is [[continuous]] (where {{M|G\times G}} is considered with the [[product # {{M|i:G\rightarrow G}} with {{M|i:x\rightarrow x^{-1} }} is also continuous
  1 KB (224 words) - 06:23, 8 August 2016
• Notes:Tangent space
  * {{M|1=M:=\{p\in\mathbb{R}^3\ \vert\ p=(x,y,x^2+y^2)\}\subset\mathbb{R}^3}} (with the subspace topology) and the two charts: *# {{M|1=i:\mathbb{R}^2\rightarrow M}} by {{M|1=i:(x,y)\mapsto(x,y,x^2+y^2)}} (think of it as like the "identity chart" it is what the manifold is)
  5 KB (1,002 words) - 19:42, 15 August 2016
• Notes:Basis for a topology/McCarty
  Let {{Top.|X|J}} be a [[topological space]], let {{M|\mathcal{B}\subseteq\mathcal{J} }}. Suppose {{M|\mathcal{K} }} is another [[topology]] on {{M|X}} and {{M|\mathcal{B}\subseteq\mathcal{K} }}, then:
  3 KB (467 words) - 16:58, 16 August 2016
• Topological vector space
  ...}} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.|X|J}} is a [[topological space]]. We call the [[tuple]]: ...ogical vector space". The topology is "more implicit" when we speak of {{M|X}} than the field of a vector space is, so often we will just write:
  2 KB (383 words) - 14:03, 16 February 2017
• Doctrine:Measure theory terminology/Proposals
  ...\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} we call a set, {{M|X\in\mathcal{H} }}, {{M|\mu^*}}-measurable if: ** {{M|1=\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)]}}
  3 KB (466 words) - 21:29, 20 August 2016
• Outer splicing set
  ...arrow\overline{\mathbb{R}_{\ge 0} } }} be an [[outer-measure]]. A set, {{M|X\in\mathcal{H} }}, is said to be an ''outer splicing set''<ref group="Note"> * {{M|1=\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)]}}<ref group="Note">Some authors, for example Halmos, abuse notation quite
  2 KB (378 words) - 22:09, 20 August 2016
• Formula (FOL)
  # If {{M|A}} is a formula and {{M|x}} is a {{link|variable symbol|FOL}} then both: # {{M|\exists x(x\doteq y)}} is easier to read than {{M|\exists xx\doteq y}}
  3 KB (500 words) - 05:37, 8 September 2016
• Semantics of logical connectives (FOL)
  Suppose {{M|X}} and {{M|Y}} are variables which may only take on truth values (eg, are the [[semantic ! style="min-width:3em;" | {{M|\mathbf{X} }}
  2 KB (369 words) - 07:46, 11 September 2016
• Notes:Quotient topology/Table
  ...on {{M|X'}}, {{M|\mathcal{K} }} is defined as: {{M|\forall U\in\mathcal{P}(X')[U\in\mathcal{K}\iff v^{-1}(U)\in\mathcal{J}]}} ...n\mathcal{P}(Y)}} is [[open set|open]] {{iff}} {{M|f^{-1}(U)}} open in {{M|X}}.
  2 KB (327 words) - 16:09, 13 September 2016
• HomotopyPage
  ...So simply by exhibiting a continuous function, {{M|F:X\times I\rightarrow Y}}, we get homotopic maps. So a homotopy warrants a definition, even if it i ...l space|s}} and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. A ''homotopy, relative to {{M|A}}'' is, in its purest form, is any ''[[c
  2 KB (401 words) - 12:53, 15 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
  ...l spaces]]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. Then: ...ap]], {{M|f:X\rightarrow Y}} to another continuous map, {{M|g:X\rightarrow Y}}, if {{M|f}} and {{M|g}} are [[homotopic]] is an equivalence relation{{rAI
  2 KB (272 words) - 23:37, 14 October 2016
• Notes:Homotopy terminology/Terminology
  ...{{M|(:X\rightarrow Y)}}. Let {{M|1=I:=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}\subset\mathbb{R} }} # '''Homotopy - ''' any continuous map of the form {{M|H:X\times I\rightarrow Y}} such that: {{M|1=\forall a\in A\forall s,t\in I[H(a,t)=H(a,s)]}}.
  900 B (184 words) - 14:40, 16 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces/proof
  # For all {{M|f\in C^0(X,Y)}} that {{M|f\simeq f\ (\text{rel }A)}}, symbolically: #* [[Reflexive]]: {{M|1=\forall f\in C^0(X,Y)[\homo{f}{f}]}}
  3 KB (533 words) - 07:33, 18 September 2016
• Characteristic property of the disjoint union topology/Statement
  ...}} family) and lastly, let {{M|f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y}} be a [[map]] (not necessarily [[continuous]]) then:<br/> ...all\alpha\in I\big[f\big\vert_{X_\alpha^*}:i_\alpha({X_\alpha})\rightarrow Y\text{ is continuous}\big]}}
  1 KB (238 words) - 20:05, 25 September 2016
• Characteristic property of the subspace topology/Statement
  ...m>\xymatrix{ Y \ar[r]^f \ar[dr]_{i_S\circ f} & S \ar@{^{(}->}[d]^{i_S}\\ & X}</m></center> ...}}<ref group="Note">This means {{M|S\in\mathcal{P}(X)}}, or {{M|S\subseteq X}} of course</ref>. The ''characteristic property of the subspace topology''
  2 KB (262 words) - 22:45, 25 September 2016
• The composition of continuous maps is continuous
  ...aces]] (not necessarily distinct) and let {{M|f:X\rightarrow Y}} and {{M|g:Y\rightarrow Z}} be ''[[continuous]]'' [[maps]], then{{rITTMJML}}: ...n|map}}, {{M|g\circ f:X\rightarrow Z}}, given by {{M|g\circ f:x\mapsto g(f(x))}}, is a continuous map.
  823 B (137 words) - 23:14, 25 September 2016
• Doctrine:Homotopy terminology
  * Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] * Let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset]] of {{M|X}}
  3 KB (535 words) - 09:01, 31 October 2016
• Every injection yields a bijection onto its image
  ...{f}}</m>Let {{M|X}} and {{M|Y}} be [[sets]] and suppose {{M|f:X\rightarrow Y}} is any ''[[injective]]'' [[map]] between them. Then we claim that there i ...f}:x\mapsto f(x)}}- where {{M|f(X)}} denotes the {{link|image|map}} of {{M|X}} under {{M|f}}<ref name="imageDef" group="Note" /><!--body of note declare
  4 KB (813 words) - 11:53, 26 September 2016
• Every bijection yields an inverse function
  Let {{M|X}} and {{M|Y}} be [[sets]] and suppose {{M|f:X\rightarrow Y}} is a [[map]] between them, and that it is a [[bijective]] map. Then there * {{M|f^{-1}:Y\rightarrow X}} such that {{M|1=f^{-1}(y)=x\iff f(x)=y}}
  588 B (100 words) - 12:24, 26 September 2016
• Open map
  ...map]] (not necessarily [[continuous]] - just a map between {{M|X}} and {{M|Y}} considered as sets), then {{nowrap|we call {{M|f}} an ''open map'' if{{rI ...all {{plural|open set|s}} of {{Top.|X|J}} are [[open set|open]] in {{Top.|Y|K}}
  664 B (118 words) - 22:53, 22 February 2017
• Closed map
  ...map]] (not necessarily [[continuous]] - just a map between {{M|X}} and {{M|Y}} considered as sets), then {{nowrap|we call {{M|f}} a ''closed map'' if{{r ...{plural|closed set|s}} of {{Top.|X|J}} are [[closed set|closed]] in {{Top.|Y|K}}
  1 KB (246 words) - 19:59, 26 September 2016
• Task:Equivalent properties to homeomorphism
  ...|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]]. Then{{rITTMJML}}:
  417 B (67 words) - 20:25, 26 September 2016
• Every surjective map gives rise to an equivalence relation
  ...operty is never used! It is true though that every map, {{M|f:X\rightarrow Y}} gives rise to an equivalence relation, where {{M|x_1\sim x_2}} if {{M|1=f ...case we can {{link|factor|function}} {{M|f}} through {{M|\pi:X\rightarrow X/\sim}} always to yield {{M|\bar{f} }}, and "distil" the information of {{M|
  2 KB (315 words) - 13:54, 8 October 2016
• Dense
  ...{{M|X}}, say {{M|A\in\mathcal{P}(X)}}, we say {{M|A}} is ''dense'' in {{M|X}} if: ...M|\text{Closure}(A)}}]]{{M|\eq X}} (sometimes written: {{M|\overline{A}\eq X}})
  6 KB (1,097 words) - 04:15, 1 January 2017
• Equivalent statements to a set being dense
  ...rary [[subset of]] {{M|X}}. Then "{{M|E}} is [[dense set|dense]] in {{Top.|X|J}}" is equivalent to any of the following: # {{M|1=\forall U\in\mathcal{J}[U\ne\emptyset\implies\exists y\in E[y\in U]]}}
  3 KB (490 words) - 20:18, 28 October 2016
• Every continuous map from a non-empty connected space to a discrete space is constant
  ...nk|connected|topology}}'' [[topological space]] and let {{M|(Y,\mathcal{P}(Y))}} be any [[discrete topology|discrete topological space]], then{{rITTMJML ...It should go without saying, but a map is constant if {{M|1=\forall p,q\in X[f(p)=f(q)]}}</ref>
  851 B (138 words) - 23:02, 30 September 2016
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
  ...M|1=\mathcal{K}:=\mathcal{P}(Y)}}<ref group="Note">Note: {{M|1=\mathcal{P}(Y)=\mathcal{P}(\{0,1\})=\{\emptyset,\{0\},\{1\},\{0,1\}\} }}</ref> then{{rITT * {{Top.|X|J}} is {{link|disconnected|topology}} (ie, not {{link|connected|topology}})
  1 KB (172 words) - 23:12, 30 September 2016
• The image of a connected set is connected
  ...row Y}} be a ''[[continuous]]'' [[map]]. Then, for any {{M|A\in\mathcal{P}(X)}}, we have{{rITTBM}}: ...t|topology}} of {{Top.|X|J}} then {{M|f(A)}} is connected subset in {{Top.|Y|K}}
  2 KB (422 words) - 04:10, 3 October 2016
• Notes:Homology
  & & x \ar@/_.25pc/[dll]_a \\ y \ar@/_.25pc/[drr]_b \\
  6 KB (897 words) - 07:30, 15 October 2016
• Fibre
  Let {{M|f:X\rightarrow Y}} be a [[function]], then{{rITTMJML}}: ...'' of {{M|f}} is any set of the form {{M|f^{-1}(\{y\})}} for some {{M|y\in Y}}
  735 B (128 words) - 12:59, 16 October 2016
• Exercises:Mond - Topology - 1/Question 6
  ** '''Note: ''' The reason for the odd choice of {{M|\sin}} for the {{M|x}} coordinate, and the minus signs is because my first choice was <span styl # we get {{M|\pi:[-1,1]\rightarrow\frac{[-1,1]}{\sim} }}, {{M|\pi:x\mapsto [x]}} automatically and it is continuous.
  7 KB (1,326 words) - 12:26, 12 October 2016
• The image of a compact set is compact
  ...l{P}(X)}} be an arbitrary [[subset of]] {{M|X}} and let {{M|f:X\rightarrow Y}} be a ''[[continuous]]'' [[map]]. Then: ...}} being an [[open cover]] of {{M|A}} by sets [[open set|open]] in {{Top.|X|J}}
  2 KB (332 words) - 17:20, 18 December 2016
• Exercises:Mond - Topology - 1/Question 7
  ...}^3}}, it should be clear that for all {{M|x\in H-\partial H}} that {{M|f'(x)}} is intended to be a point on the red sphere and that {{M|1=f'\big\vert_{ ...pi:x\mapsto [x]}} where {{M|[x]}} denotes the [[equivalence class]] of {{M|x}}
  9 KB (1,732 words) - 23:26, 11 October 2016
• Homeomorphic
  Two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}}, are said to be ''homeomorphic'' if there exists a [[homeomorphism]] be ...the morphisms (which are continuous maps) between {{Top.|X|J}} and {{Top.|Y|K}}.
  883 B (132 words) - 11:52, 8 October 2016
• Equivalence relation induced by a function
  ...hen {{M|f}} induces an [[equivalence relation]], {{M|\sim\subseteq X\times X}} where<ref>[[File:MondTop2016ex1.pdf]]</ref>: * for {{M|x_1,x_2\in X}} we say {{M|x_1\sim x_2}} if {{M|1=f(x_1)=f(x_2)}}
  2 KB (276 words) - 22:40, 8 October 2016
• Passing to the quotient (function)/Diagram
  X \ar[r]^w \ar[dr]_f & W \ar@{.>}[d]^{\tilde{f}}\\ & Y
  248 B (40 words) - 20:38, 8 October 2016
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection/Diagram
  ...-size:1.5em;" | <center><m>\xymatrix{ X \ar[r]^f \ar[d]_{\pi} & Y \\ \frac{X}{\sim} \ar@{.>}[ur]_{\tilde{f} } }</m></center>
  261 B (38 words) - 22:54, 8 October 2016
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
  ...ow Y}} be any [[function]] between them, and let {{M|\sim\subseteq X\times X}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by * {{M|1=\forall x,x'\in X[x\sim x'\iff f(x)=f(x')]}}
  6 KB (1,097 words) - 20:24, 9 October 2016
• If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection
  ...urjective]]'' [[function]] between them, and let {{M|\sim\subseteq X\times X}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by * {{M|1=\forall x,x'\in X[x\sim x'\iff f(x)=f(x')]}}
  2 KB (339 words) - 12:57, 9 October 2016
• Characteristic property of the quotient topology/Statement
  ...|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|q:X\rightarrow Y}} be a {{link|quotient map|topology}}. Then{{rITTMJML}}: * For any topological space, {{Top.|Z|H}} a [[map]], {{M|f:Y\rightarrow Z}} is [[continuous]] {{iff}} the composite map, {{M|f\circ q}},
  495 B (71 words) - 22:16, 9 October 2016
• Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map
  ..., but in addition the map it yields, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}}, is a continuous [[injection]]<ref>[[File:MondTop2016ex1.pdf]]</ref>. ...row Y}} is [[surjective]] then so is {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} also, making {{M|\bar{f} }} a [[bijection]]<ref group="Note">See: ''[[If
  3 KB (430 words) - 22:23, 9 October 2016
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
  ...s inverse was also continuous</ref>, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}}<ref>[[File:MondTop2016ex1.pdf]]</ref>. ...n injective continuous map]]''" that {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} is [[injective]] and [[continuous]].
  2 KB (264 words) - 22:32, 9 October 2016
• Exercises:Mond - Topology - 1/Question 8
  ...on]]'' [[equivalence relation induced by a map|induced by {{M|f}}]] on {{M|X}}. ...Y}} is [[Hausdorff]] then {{M|\frac{X}{\sim} }} is [[homeomorphic]] to {{M|Y}}.
  3 KB (413 words) - 00:13, 12 October 2016
• Exercises:Mond - Topology - 1/Question 9
  ...]] that defines (for {{M|x\in\mathbb{S}^2\subset\mathbb{R}^3}}) {{M|x\sim -x}}, that is it identifies antipodal points. Consider now {{M|1=\{x,-x\}=\pi^{-1}(a)}} and {{M|1=\{y,-y\}=\pi^{-1}(b)}}:
  8 KB (1,450 words) - 12:34, 12 October 2016
• Properties of the pre-image of a function
  Let {{M|X}} and {{M|Y}} be [[sets]] and let {{M|f:X\rightarrow Y}} be a [[function]] between them. Then: # For {{M|1=\{A_\alpha\}_{\alpha\in I}\subseteq\mathcal{P}(Y)[f^{-1}(\bigcup_{\alpha\in I}A_\alpha)=\bigcup_{\alpha\in I}f^{-1}(A_\alpha
  2 KB (417 words) - 00:51, 14 October 2016
• A map is continuous if and only if the pre-image of every closed set is closed
  ...|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[mapping]], then{{rITTMJML}}{{rITTBM}}{{rTJRM}}: * {{M|f:X\rightarrow Y}} is [[continuous]] {{iff}} {{M|1=\forall E\in C(\mathcal{K})[f^{-1}(E)\in
  2 KB (378 words) - 01:39, 14 October 2016
• A map is continuous if and only if each point in the domain has an open neighbourhood for which the restriction of the map is continuous on
  ...J}} and {{Top.|Y|K}} be [[topological spaces]], and let {{M|f:X\rightarrow Y}} be a [[map]] between them, then{{rITTMJML}}: ...riction|function}} of {{M|f}} to {{M|U}}, so {{M|f\big\vert_U:U\rightarrow Y}} by {{M|1=f\big\vert_U:u\mapsto f(u)}}</ref> is [[continuous]]
  4 KB (710 words) - 06:01, 14 October 2016
• Pasting lemma
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|\{A_\alpha\}_{\alpha\in I} }} be eit # An arbitrary [[open cover]] of {{M|X}}, or
  1 KB (193 words) - 07:07, 14 October 2016
• Definitions and iff
  * An {{M|X}} is {{M|D}} when it satisfies {{M|P(X)}} (for some statement {{M|P}}), symbolically: ** {{M|1=\forall X[P(X)\implies D]}}
  1 KB (202 words) - 00:14, 15 October 2016
• Saturated set with respect to a function
  ...ny [[function]] between them. A [[subset of]] {{M|X}}, {{M|U\in\mathcal{P}(X)}}, is said to be ''saturated with respect to {{M|f}}'' if{{rITTMJML}}: * {{M|1=\exists V\in\mathcal{P}(Y)[U=f^{-1}(V)]}}, in words:
  785 B (138 words) - 13:08, 16 October 2016
• Equivalent conditions to a set being saturated with respect to a function
  ...function]]. Let {{M|U\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}, then{{rITTMJML}}: *# if {{M|x\in U}} then every point {{M|x'\in X}} such that {{M|1=q(x)=q(x')}} is also in {{M|U}}<ref name="ITTMJML"/>
  807 B (142 words) - 13:15, 16 October 2016
• Equivalent conditions to a map being a quotient map
  ...|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]]. Then{{rITTMJML}}:
  637 B (105 words) - 13:38, 16 October 2016
• Exercises:Mond - Topology - 2/Section B/Question 5
  ...2\subset\mathbb{R}^2}} where {{M|1=I:=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}\subset\mathbb{R} }} with {{M|1=v_0=(0,0)}}, {{M|1=v_1=(1,0)}}, {{M|1 ...)=\pi(v_0)}}) contains a point inside and a point outside the image of {{M|X}} (green). To find this we use open balls in {{M|I^2}} contained inside the
  8 KB (1,427 words) - 08:30, 18 October 2016
• Module
  * left action, {{M|[:R\times M\rightarrow M]}} given by {{M|[:(r,x)\mapsto rx]}} of {{M|R}} on {{M|M}}, called the "left {{M|R}}-module struct # {{M|1=\forall r,s\in R,\forall x\in M[r(sx)=(rs)x]}},
  1 KB (246 words) - 22:40, 19 October 2016
• Module homomorphism
  *# {{M|1=\forall x,y\in M[\varphi(x+y)=\varphi(x)+\varphi(y)]}} and *# {{M|1=\forall r\in R,\forall x\in M[\varphi(rx)=r\varphi(x)]}}<ref group="Note">A homomorphism of right modules is the same but this r
  3 KB (432 words) - 22:20, 19 October 2016
• Notes:Chain complex of modules
  We denote the ''homology class'' of {{M|x\in\text{Ker}(\partial_n)}} by {{M|1=\text{cls }z:=z+\text{Im}(\partial_{n+1 ...ain transformation {{M|\mathcal{C}(f):\mathcal{C}(X)\rightarrow\mathcal{C}(Y)}} of their singular chain complexes.
  3 KB (440 words) - 17:15, 20 October 2016
• Quotient module
  ...{A}\times\frac{M}{A}\rightarrow\frac{M}{A} }} by {{M|+:([x],[y])\mapsto [x+y]}} ...'' {{M|\cdot:R\times\frac{M}{A}\rightarrow\frac{M}{A} }} by {{M|\cdot:(r,[x])\mapsto [rx]}}
  1 KB (209 words) - 20:00, 23 October 2016
• Notes:Coset stuff
  ...must be this way as we will require {{M|x\sim x}}, then we get {{M|1=x^{-1}x=e\in H}} as {{M|H}} is a subgroup.</ref>. Done - [[User:Alec|Alec]] ([[User #* By symmetry this is/must be the same as {{M|y^{-1}x\in H}} - this is true as {{M|H}} is a subgroup.
  6 KB (1,212 words) - 21:17, 25 October 2016
• Notes:Rings and u-rings
  *# {{M|1=f(x+y)=f(x)+f(y)}} and *# {{M|1=f(xy)=f(x)f(y)}}
  527 B (92 words) - 16:33, 26 October 2016
• The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set
  ...[[intersection is commutative]], it follows that {{M|1=\iff(\exists y\in B[y\in A]}} of course, as if we applied the claim to {{M|B\cap A}} instead. * Suppose {{M|1=\exists x\in A[x\in B]}}
  2 KB (287 words) - 19:31, 28 October 2016
• The set of continuous functions between topological spaces
  ...[functions]] from {{M|X}} to {{M|Y}}, with respect to the {{plural|topolog|y|ies}}: {{M|\mathcal{J} }} and {{M|\mathcal{K} }}. * {{M|\big(f\in C(X,Y)\big)\iff\big(f:X\rightarrow Y\text{ is a continuous function}\big)}}
  1 KB (235 words) - 05:02, 3 November 2016
• Index of spaces, sets and classes
  * [[The set of continuous functions between topological spaces]] - {{M|C(X,Y)}} ** [[C(I,X)|{{M|C([0,1],X)}}]]
  335 B (61 words) - 05:06, 3 November 2016
• Homotopy concatenation
  ...osed unit interval]]</ref> be {{plural|homotop|y|ies}} from {{M|X}} to {{M|Y}}. Suppose: * {{M|1=\forall x\in X[H_1(x,1)=H_2(x,0)]}} - that the final stage of {{M|H_1}} is the same as the initial stage
  2 KB (260 words) - 05:09, 6 November 2016
• Homotopy invariance of loop concatenation
  ...is a path and {{M|1=\ell(0)=\ell(1)=b}}.<br/><br/>Furthermore, {{M|\Omega(X,b)}} is not just a [[set]], it does have a [[group]] structure we can imbue * {{M|1=\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)[([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2'])\implies[\ell_1*\ell_2]=[\el
  3 KB (462 words) - 09:21, 6 November 2016
• Homotopy invariance of path concatenation
  ...H_1,H_2:[0,1]\times [0,1]\rightarrow X}} are the specific {{plural|homotop|y|ies}} of the {{link|path|topology|s}}) then{{rITTMJML}}: *** {{M|1=p_1*p_2:[0,1]\rightarrow X}} by {{M|p_1*p_2:t\mapsto\left\{\begin{array}{lr}p_1(2t)&\text{for }t\in[0,
  2 KB (273 words) - 19:11, 9 November 2016
• Notes:Fundamentals of Algebraic Topology - Weintraub
  ...al{P}(X)}}. We let {{M|f:X\rightarrow Y}} denote a map from {{M|X}} to {{M|Y}} as usual. ...|1=f\big\vert_A:A\rightarrow B}}; or more simply, a map {{M|f:X\rightarrow Y}} with {{M|f(A)\subseteq B}}
  2 KB (325 words) - 21:53, 8 November 2016
• Doctrine:Differentiation notation & terminology
  ...{M|(Y,\Vert\cdot\Vert_Y)}} be [[normed spaces]] and let {{M|f:X\rightarrow Y}} be a [[function]]. Then: * {{M|f}} is ''[[differentiable]]'' at {{M|a\in X}} if:
  3 KB (628 words) - 10:34, 11 November 2016
• Quotient by an equivalence relation
  ...[[quotient]]'' of {{M|X}} by {{M|\sim}}, denoted {{M|X/\sim}} or {{M|\frac{X}{\sim} }} is defined as follows: ...vert\ x\in X\} }} where {{M|[x]}} denotes the [[equivalence class]] of {{M|x}}.
  2 KB (295 words) - 14:16, 13 November 2016
• Equivalence classes are either equal or disjoint
  ...lastly let {{M|\pi:X\rightarrow\frac{X}{\sim} }} given by {{M|\pi:x\mapsto[x]}} be the [[canonical projection of the equivalence relation]]. Then: ...that {{M|\frac{X}{\sim} }} is a {{link|partition|abstract algebra}} of {{M|X}}. That is:
  2 KB (388 words) - 14:36, 13 November 2016
• Absolute value (object)
  ..._v:F\rightarrow\mathbb{R} }} given by {{M|\vert\cdot\vert_v:x\mapsto \vert x\vert_v}} # {{M|\forall x\in F[\vert x\vert_v\ge 0]}}
  2 KB (350 words) - 05:25, 21 November 2016
• Real sequence
  ...\vert}} - the [[absolute value]] of the difference between {{M|x}} and {{M|y}}. We can subtract {{M|y}} from {{M|x}} as the reals are a [[field]].
  683 B (115 words) - 05:44, 23 November 2016
• Operations on convergent sequences of real numbers
  ...considered convergent with respect to the [[metric]] {{M|d(x,y):\eq\vert x-y\vert}}). Suppose that [[limit (sequence)|{{M|\lim_{n\rightarrow\infty}(a_n) ***** Noting that {{M|d(x,y):\eq \vert x-y\vert}} we see
  5 KB (900 words) - 05:45, 23 November 2016
• The vector space of all linear maps between two spaces
  ..., here {{M|\mathcal{F}(U,V)}} denotes the [[set of all functions from X to Y|set of all functions from {{M|U}} to {{M|V}}]]<ref group="Note">You may hav * [[The set of all continuous maps between spaces]] - {{M|C(X,Y)}}
  2 KB (400 words) - 21:16, 17 November 2016
• Example:A smooth function that is not real analytic
  * {{MM|1=f:x\mapsto\left\{\begin{array}{lr}e^{-\frac{1}{x} } & \text{if }x>0\\ 0 & \text{otherwise}\end{array}\right.}} ===For {{M|x<0}}, {{M|f}} is smooth===
  5 KB (1,030 words) - 04:25, 27 November 2016
• Weierstrass approximation theorem
  ...]] from the [[closed interval]] {{M|[a,b]:\eq\{x\in\mathbb{R}\ \vert\ a\le x\le b\}\subset\mathbb{R} }} to the [[real line]], {{M|\mathbb{R} }}. We cons ...M|\Vert\cdot\Vert_\infty:f\mapsto\mathop{\text{Sup} }_{x\in [a,b]}(\vert f(x)\vert)}} where {{M|\vert\cdot\vert:\mathbb{R}\rightarrow\mathbb{R} }} is, a
  8 KB (1,610 words) - 08:17, 28 December 2016
• Notes:Free group
  * Let {{M|X}} be a [[set]]. * Let {{M|X'}} be a [[disjoint]] set
  3 KB (544 words) - 20:36, 10 December 2016
• Nth homotopy group
  : '''Note: ''' the [[fundamental group]] is {{M|\pi_1(X,p)}} ...any fixed point. The {{M|n^\text{th} }} homotopy group, written {{M|\pi_n(X,x_0)}} is defined as follows:
  2 KB (409 words) - 22:17, 12 December 2016
• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
  ...]] and let {{M|f_1,f_2:I\rightarrow X}} be {{link|path|topology|s}} in {{M|X}} such that: ...e will do this by constructing a [[homotopy]], {{M|H':I\times I\rightarrow Y}}, between them. We know that:
  2 KB (282 words) - 22:38, 12 December 2016
• A continuous map induces a homomorphism on fundamental groups
  ...base point for the [[fundamental group]] of {{M|X}} at {{M|p}}, {{M|\pi_1(X,p)}}. Then{{rITTMJML}}: ...oup homomorphism|homomorphism of the fundamental groups of {{M|X}} and {{M|Y}}]]
  8 KB (1,475 words) - 07:35, 14 December 2016
• Fundamental group homomorphism induced by a continuous map
  ...he fundamental group]], {{M|\pi_1(X,p)}}) and let {{M|\varphi:X\rightarrow Y}} be a [[continuous map]]. Then{{rITTMJML}}: ...up homomorphism]] on the fundamental groups, {{M|\pi_1(X,p)}} to {{M|\pi_1(Y,\varphi(p))}}, which we denote:
  3 KB (442 words) - 04:28, 14 December 2016
• The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms/Statement
  ...roup]] {{M|\pi_1(X,p)}}) and let {{M|\varphi:X\rightarrow Y}} and {{M|\psi:Y\rightarrow Z}} be [[continuous maps]]. Then{{rITTMJML}}: ...p|fundamental group homomorphism, {{M|\varphi_*:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))}}, induced by {{M|\varphi}}]] - and "" for the others
  841 B (133 words) - 05:01, 14 December 2016
• Homeomorphic topological spaces have isomorphic fundamental groups/Statement
  ...the fundamental group]] {{M|\pi_1(X,p)}}) and let {{M|\varphi:X\rightarrow Y}} be that homeomorphism. Then: {{rITTMJML}}: * {{M|\pi_1(X,p)\cong\pi_1(Y,\varphi(p))}} - where {{M|\cong}} denotes [[group isomorphism]] here, but c
  707 B (110 words) - 05:56, 14 December 2016
• If the composition of two functions is a bijection then the initial map is injective and the latter map is surjective
  ...ow Y}} and {{M|g:Y\rightarrow Z}} be [[maps]]. Suppose that {{M|(g\circ f):X\rightarrow Z}} is a [[bijection]]. Then{{rITTMJML}}: # {{M|f:X\rightarrow Y}} is an [[injection]], and
  2 KB (462 words) - 08:27, 14 December 2016
• A function is continuous if and only if the pre-image of every basis element is open
  ...a [[topological basis|basis]] of {{Top.|Y|K}}, and let {{M|f:X\rightarrow Y}} be a [[mapping]] between them<ref group="Note">We do not assume anything * {{M|f:X\rightarrow Y}} is [[continuous]] {{iff}} {{M|\forall B\in\mathcal{B}[f^{-1}(B)\in\mathca
  4 KB (839 words) - 18:35, 17 December 2016
• Notes:Concrete tensor product
  ...ions of the form {{M|(:X\rightarrow Y)}} then {{M|\forall x\in X[f(x)\eq g(x)]}} - we must do this.
  3 KB (537 words) - 21:04, 21 December 2016
• Index of notation for sets of continuous maps/Index
  ...Y)}}]] - for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}, {{M|C(X,Y)}} is the [[set]] of all [[continuous maps]] between them. ...}}, set of all {{link|path|topology|s}} on a [[topological space]] {{Top.|X|J}}
  2 KB (463 words) - 06:20, 1 January 2017
• Index of notation/C
  ! [[C(X,Y)|{{M|C(X,Y)}}]] ...example: [[C(I,X)|{{M|C(I,X)}}]] - all {{link|path|topology|s}} in {{Top.|X|J}}. These sets often have additional structure (eg, [[vector space]], [[al
  958 B (151 words) - 06:13, 1 January 2017
• Order terminology
  If {{M|(X,\preceq)}} is a [[poset]], a set with a [[partial order]], then we get a [[ * {{M|x\prec y\iff(x\preceq y\wedge x\neq y)}}
  4 KB (656 words) - 18:45, 8 January 2017
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.5
  ...mathbb{Z}^3\ \vert\ \exists x\in\mathbb{Z}^3[A(x)\eq y]\}\eq\{A(x)\ \vert\ x\in\mathbb{Z}^3\} }} ** {{M|\text{Ker}(A):\eq\{x\in\mathbb{Z}^3 \ \vert\ A(x)\eq 0\} }}
  3 KB (473 words) - 19:34, 14 January 2017
• If the intersection of two open balls is non-empty then for every point in the intersection there is an open ball containing it in the intersection
  ...that {{M|B_{r_1}(x_1)}} and {{M|B_{r_2}(x_2)}} are [[open balls]] of {{M|(X,d)}}). Let {{M|B_i:\eq B_{r_i}(x_i)}} (for brevity<ref group="Note">The len ...tyset]\implies\big[\forall x_3\in B_1\cap B_2\exists r_3\in\mathbb{R}_{>0}[x\in B_3\wedge B_3\subseteq B_1\cap B_2]\big]}}<ref group="Note">{{XXX|The fo
  6 KB (1,007 words) - 20:16, 16 January 2017
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.2
  ...} }}{{M|\underline{\text{G} } }}{{M|\sf{, } }}{{M|\text{J} }}{{M|\sf{, T, Y} \} }}</span> * <span style="font-size:1.5em;">{{M|\{\sf{k, X, } }}{{M|\mathcal{Z}\} }}</span>
  17 KB (3,132 words) - 12:03, 18 January 2017
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.2/Lemmas
  ...[[homeomorphism]] between them (so {{M|X\cong Y}}), then for any {{M|x\in X}} we have: ...\} }:X-\{x\}\rightarrow Y}} - notice the codomain has changed to {{M|Y-\{f(x)\} }}</ref> we of course consider these spaces with the [[subspace topology
  2 KB (388 words) - 18:46, 17 January 2017
• Notes:CW-Complex
  ...]]'' [[open cell|open cells]], {{M|\{e_\alpha\}_{\alpha\in I} }}, with {{M|X\eq\bigcup_{\alpha\in I}e_\alpha}}, such that: # {{Top.|X|J}} is a [[Hausdorff space]]
  10 KB (1,736 words) - 01:00, 23 January 2017
• Notes:Delta complex/Formal attempt
  ...ures that it is 1-to-1. We can never have equality of {{M|f(x)}} and {{M|f(y)}}
  5 KB (966 words) - 14:36, 6 February 2017
• Convex set
  ...} or the [[complex numbers]], {{M\mathbb{C} }} and let {{M|C\in\mathcal{P}(X)}} be given. Then we say {{M|C}} is ''convex'' if: * {{M|\forall x,y\in C\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in C]}}
  799 B (143 words) - 11:27, 9 February 2017
• Convex
  ...ce]] {{M|(X,\mathbb{K})}} a [[subset of]] {{M|X}}, say {{M|C\in\mathcal{P}(X)}} is convex if ** {{M|\forall x,y\in C\forall t\in[0,1]\subset\mathbb{R}[x+t(y-x)\in C]}} - the line between any two points in the set is also in the set
  532 B (92 words) - 14:50, 9 February 2017
• Notes:Functional Analysis II
  ** He uses {{M|B_X(0,1)}} - ball notation for {{M|(X,\Vert\cdot\Vert)}} centred at {{M|0}} of radius {{M|1}} - {{Caveat|easily m ...- Let {{M|S\in\mathcal{P}(X)}} be given. Symmetric if {{M|\forall x\in S[-x\in S]}}
  4 KB (818 words) - 12:00, 9 February 2017
• Convex function
  #* In symbols: {{M|\forall x,y\in S\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in S]}}, and ...y]}} is {{M|\le}} the point {{M|t}}-far along the line {{M|f(x)}} to {{M|f(y)}}
  1 KB (224 words) - 10:54, 10 February 2017
• Epigraph
  ...|G:\eq\{ (x_1,\ldots,x_n,y)\in\mathbb{R}^{n+1}\ \vert\ x\in S\wedge y\eq f(x)\} }}
  674 B (114 words) - 11:04, 10 February 2017
• Exercises:Saul - Algebraic Topology - 5/Exercise 5.6
  ...retraction being {{M|r:X\rightarrow A}}). Lastly take i {{M|i:A\rightarrow X}} to be the [[inclusion map]], {{M|i:a\mapsto a}}. Show that: {{M|H_*^s(X)\cong H_*^s(A)\oplus H_*^s(X,A)}}
  1 KB (269 words) - 20:13, 14 February 2017
• For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal
  ...ctor space]] and let {{M|(Y,\mathbb{K})}} be a [[vector subspace]] of {{M|(X,\mathbb{K})}}, then{{rFAVIDMH}}: * {{M|(\exists U\in(\mathcal{J}-\{\emptyset\})[U\subseteq Y])\implies X\eq Y}}
  924 B (140 words) - 17:51, 16 February 2017
• A proper vector subspace of a topological vector space has no interior
  ...{K})}} be a [[vector subspace]] of {{M|(X,\mathbb{K})}} (so {{M|Y\subseteq X}}), then{{rFAVIDMH}}: ...r subspace]] of {{M|X}} then [[Interior (topology)|{{M|\text{Int} }}]]{{M|(Y)\eq\emptyset}} - {{ie}} it has no interior, or empty interior.
  2 KB (415 words) - 17:49, 16 February 2017
• Trivial
  ...ent is [[vacuous]], for example {{M|\forall x\in X[P(X)]}} holds, when {{M|X\eq\emptyset}} is it ''vacuously true''. ...ivial metric, {{M|d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x\eq y\\ 1&\text{otherwise}\end{array}\right.}}, [[topology induced by a metric|in
  1 KB (171 words) - 13:29, 16 February 2017
• Interior (topology)
  ...ry [[subset of]] {{M|X}}, the ''interior'' of {{M|A}}, with respect to {{M|X}}, is denoted and defined as follows{{rITTMJML}}: ...e considering the interior of {{M|A}} with respect to the open sets of {{M|X}}.
  2 KB (328 words) - 20:10, 16 February 2017
• Local homeomorphism
  ...|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] (we do not require [[continuity]] at this stage). We call {{ * {{M|\forall x\in X\exists U\in\mathcal{O}(x,X)\big[\big(f(U)\in\mathcal{K}\big)\wedge \big(f\vert_U^\text{Im}:U\rightarro
  2 KB (271 words) - 21:45, 22 February 2017
• Example:Continuous maps contracting something to a point
  File:ExampleGraphX1.JPG|{{Top.|X|J}} File:ExampleGraphX2.JPG|{{Top.|Y|K}}
  700 B (103 words) - 14:35, 23 February 2017
• Notes:Covering spaces
  ...t {{M|p:E\rightarrow X}} be [[continuous map]], we say {{M|U}} open in {{M|X}} is ''evenly covered'' by {{M|p}} if: ** {{M|p:E\rightarrow X}} is a covering map if:
  3 KB (658 words) - 19:20, 25 February 2017
• Lifting of a continuous map through a covering map
  ...m;"><m>\xymatrix{ & & E \ar[d]_p \\ Y \ar[rr]^f \ar@{-->}[rru]^\varphi & & X }</m></span></center> ...ology}} of {{M|X}}, with {{link|covering map|topology}} {{M|p:E\rightarrow X}}. Then{{rITTMJML}}{{rITTGG}}:
  2 KB (337 words) - 01:59, 26 February 2017
• The real numbers/Infobox
  |data4={{M|\Vert x\Vert:\eq\sqrt{\langle x,x\rangle}\eq\vert x\vert}}<br/> - [[Euclidean norm]] on {{M|\mathbb{R}^1}} |data5={{M|d(x,y):\eq\Vert x-y\Vert\eq \vert x-y\vert}}<br/> - [[Absolute value]]<br/> - [[Euclidean metric]] on {{M|\mathbb
  1 KB (213 words) - 21:31, 26 February 2017
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6
  ...homology groups of {{M|T^2:\eq\mathbb{S}^1\times\mathbb{S}^1}} and of {{M|X:\eq\mathbb{S}^1\vee\mathbb{S}^1\vee\mathbb{S}^2}} # Prove that {{M|T^2}} and {{M|X}} are not [[homotopy equivalent spaces]]
  10 KB (1,664 words) - 12:43, 1 March 2017
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6/X Complex
  ! colspan="3" | {{M|\Delta}}-complex for {{M|X}} ...{\mathbf{A} } }|(.7){\LARGE{\mathbf{ B} } } & & \bullet_{v_4} \ar@{<-}[ur]^x \\
  806 B (126 words) - 16:58, 28 February 2017
• Unique lifting property
  ...e {{Top.|Y|K}} is a [[connected topological space]] and {{M|f:Y\rightarrow X}} is a [[continuous map]], then{{rITTGG}}^Partial:{{rITTMJML}}<su ...map through a covering map|lifts of {{M|f}} through {{M|p}}]], say {{M|g,h:Y\rightarrow E}} we have:
  13 KB (2,510 words) - 16:23, 2 March 2017
• Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself/Statement
  ...{M|f:X\rightarrow Y}} is a [[homeomorphism]] between them, so {{M|X\cong_f Y}}, then: * {{M|\forall A\in\mathcal{P}(X)[A\cong_{f\vert_{A}^{\text{Im} } } f(A)]}}
  708 B (124 words) - 13:24, 7 March 2017
• A pair of identical elements is a singleton
  * {{M|\exists x[x\in\{t,t\}\wedge\forall y(y\in\{t,t\}\rightarrow y\eq x)]}} (as per definition of {{link|singleton|set theory}} * {{M|\forall A\forall B\exists C\forall x(x\in C\leftrightarrow x\eq A\vee x\eq B)}} this is the pairing axiom, in this case {{M|A}} and {{M|B}} are {{M
  2 KB (315 words) - 23:35, 8 March 2017
• NeedEditingSpace
  ...\eq p(r_1):\eq p(g(t))\eq f(t)}} by {{M|g}} being a lifting, note {{M|z\in X}} ...ta\in\mathcal{K} }}, {{ie}} that {{M|W_\beta}} is [[open set|open]] in {{M|Y}}
  5 KB (881 words) - 16:24, 2 March 2017
• Axiom schema of replacement
  ...)}} - the {{link|image|function}} of {{M|X}} under {{M|F}} - denoted {{M|F(X)}} is also a set{{rSTTJ}}. ...orall y\forall z\big[\big(\varphi(x,y,p)\wedge\varphi(x,z,p)\big)\limplies y\eq z\big]}_{\text{if }(a,b)\in f\iff\varphi(a,b,p)\text{ then }f\text{ acts
  2 KB (390 words) - 15:28, 5 April 2017
• There is no set of all sets
  * {{M|\forall X\exists Y[Y\notin X]}} ** For any set {{M|X}} there exists a set {{M|Y}} such that {{M|Y\notin X}}
  2 KB (330 words) - 01:13, 9 March 2017
• Graph (topological manifold)
  ...q\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\ \big\vert\ x\in U\wedge f(x)\eq y\} }}<ref group="Note">This could surely be written: * {{M|\Gamma(f):\eq\{(x,y)\in U\times\mathbb{R}^k\ \big\vert y\eq f(x)\} }}
  2 KB (369 words) - 12:53, 17 March 2017
• Example:Smooth map between the real line and the circle (considered as smooth manifolds)
  ...}_\mathbb{R} }} is the [[identity map]], {{M|\text{Id}_\mathbb{R}:x\mapsto x}}</ref>) and let [[Circle|{{M|\mathbb{S}^1}}]] be considered as a smooth ma ...>0\} }} and {{M|\varphi_+:U^+\rightarrow I}} by {{M|\varphi_+:(x,y)\mapsto x}}
  4 KB (757 words) - 13:25, 2 April 2017
• If an inner product is non-zero then both arguments are non-zero
  Let {{M|((X,}}[[K (field)|{{M|\mathbb{K} }}]]{{M|),\langle\cdot,\cdot\rangle)}} be an [ * {{M|\forall x,y\in X[\langle x,y\rangle\neq 0\implies(x\neq 0\wedge y\neq 0)]}}
  1 KB (181 words) - 23:40, 7 April 2017
• Given a Hilbert space and a non-empty, closed and convex subset then for each point in the space there is a closest point in the subset
  Let {{M|(X,\langle\cdot,\cdot\rangle)}} be a [[Hilbert space]]<ref group="Note">Recall ...nduced by the inner-product]], {{M|\Vert\cdot\Vert:x\mapsto\sqrt{\langle x,x\rangle} }}
  3 KB (592 words) - 00:52, 7 April 2017
• K (field)
  ...e what {{M|\mathbb{K} }} means when encountered as a [[field]] (eg if {{M|(X,\mathbb{K})}} is a [[vector space]] - [[User:Alec|Alec]] ([[User talk:Alec| ...}\in\mathbb{N} }} be given and suppose {{M|X}} is a set, define {{M|z:\eq (X,\mathbb{K})}}''"
  2 KB (323 words) - 03:54, 8 April 2017
• Orthogonal complement
  ...ce]] and let {{M|L}} be a [[vector subspace]] of the [[vector space]] {{M|(X,\mathbb{K})}}<ref group="Note">{{XXX|Can we relax this to a subset maybe?}} ...ce that {{M|\langle x,y\rangle\eq 0}} is the definition of {{M|x}} and {{M|y}} being ''[[orthogonal vectors]]'', thus:
  2 KB (278 words) - 04:07, 8 April 2017
• Orthogonal vectors
  ...say {{M|x}} is ''orthogonal to'' {{M|y}} (or {{M|y}} is orthogonal to {{M|x}}) if: * {{M|\langle x,y\rangle\eq 0}}
  685 B (103 words) - 04:11, 8 April 2017
• Distance from a point to a set
  ...be given. For any point {{M|x\in X}} we define the ''distance between {{M|x}} and {{M|A}}''{{rFAVIDMH}} to be: * {{MM|d(x,A):\eq\mathop{\text{Inf} }_{a\in A}\Big(d(x,a)\Big)}}
  1,017 B (196 words) - 20:28, 9 April 2017
• The norm of a space is a uniformly continuous map with respect to the topology it induces
  ...a [[normed space]], we claim that the [[norm]] itself, {{M|\Vert\cdot\Vert:X\rightarrow\mathbb{R} }}, is a [[uniformly continuous]] map, with respect to ...X[d_{\Vert\cdot\Vert}(x,y)<\delta\implies d_\mathbb{R}(\Vert x\Vert,\Vert y\Vert)<\epsilon]}}.
  4 KB (687 words) - 20:59, 9 April 2017
• Square root (real function)
  ...}}, note that if {{M|y^2\eq x}} then {{M|-y}} is also a square root of {{M|x}}: ...(-1)^2 y^2\eq (-1)^2 x}} and that {{M|-1\times -1\eq 1}}, so {{M|(-y)^2\eq x}} also.
  2 KB (408 words) - 05:46, 10 April 2017
• Notes:Possible error 2016 compiler design q2b
  The same as mine, but with a transition via {{M|y}} from my {{M|A_5}} to {{M|G_2}}, see bottom of page 5 in [[File:Cs3250-201 ...r]^B \ar[ddr]_x \ar[r]^A \ar & *++[o][F-]{B} \ar[ur]^{\#} \ar[r]^z \ar[dr]^x & *++[o][F=]{D_1}
  2 KB (364 words) - 21:47, 9 May 2017
• Uniform continuity
  ...X,d_1)}} and {{M|(Y,d_2)}} be [[metric spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] between them. We say {{M|f}} is ''uniformly continuous'' if{ ...on>0\exists\delta>0\forall x,y\in X\big[d_1(x,y)<\delta\implies d_2(f(x),f(y))<\epsilon\big]}}
  998 B (168 words) - 21:14, 8 April 2017
• Contractible topological space
  Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is a ''contractible topological space'' if{{rITTMJML}}: * {{M|\exists c\in X\big[(:x\mapsto c)\simeq \text{Id}_X\big]}}
  3 KB (544 words) - 20:00, 24 April 2017
• Homotopy equivalent topological spaces
  ...r {{M|X}} and {{M|Y}} have the same ''homotopy type'', written {{M|X\simeq Y}}, if{{rITTMJML}}: * {{M|\exists f\in}}[[C(X,Y)|{{M|C(X,Y)}}]]{{M|\exists g\in C(Y,X)\big[(g\circ f\simeq }}[[Identity map|{{M|\text{Id}_X}}]]{{M|)\wedge(g\circ
  3 KB (596 words) - 21:13, 24 April 2017
• Graph
  ...h of {{M|f}} is: {{M|\Gamma(f):\eq\{(x,y)\in X\times Y\ \big\vert\ f(x)\eq y\} }}
  650 B (103 words) - 22:35, 4 June 2017
• Cauchy-Schwarz inequality for inner product spaces
  ...t\rangle:X\times X\rightarrow\mathbb{K} }} be an [[inner product]] so {{M|(X,\langle\cdot,\cdot\rangle)}} is an [[inner product space]], then{{rW2014LNF * {{M|\forall x,y\in X\left[\vert\ip{x,y}\vert\le\sqrt{\ip{x,x} }\sqrt{\ip{y,y} }\right]}}
  6 KB (1,279 words) - 13:00, 4 April 2017
• Poisson distribution
  |above=<span style="font-size:1.5em;">{{M|X\sim\text{Poi}(\lambda)}}</span> |data2={{MM|\mathbb{P}[X\eq k]:\eq e^{-\lambda}\frac{\lambda^k}{k!} }}
  8 KB (1,401 words) - 00:52, 20 July 2018
• Example:Joint probability distribution
  | <center><mm>\xymatrix{\mathbf{X} & \mathbf{S} & \mathbf{Y} }</mm><hr/><mm>\xymatrix{ A & 0 \ar[l] \ar[r] & C \\ & 1 \ar[ul] \ar[dr] & * {{M|X:S\rightarrow\{A,B\} }}<ref group="Note" name="rvmeasurability"/>
  7 KB (1,100 words) - 19:36, 13 September 2017
• The zero-to-the-power-of-zero problem
  ===Approach 1: {{M|x^y\eq e^{y\text{ln}(x)} }}===
  1 KB (146 words) - 21:09, 30 November 2017
• Alec's c.d.f-sidestepping sampling method
  ...fficult, impractical, impossible or otherwise unavailable, but where {{M|f(x)}} itself is easy to compute. ...0]\big]}}, there exists a [[closed interval]], {{M|[a,b]}} for which {{M|f(x)>0}} on over, or:
  3 KB (484 words) - 00:18, 1 October 2017
• Binomial distribution
  |above=<span style="font-size:1.5em;">{{M|X\sim\text{Bin}(n,p)}}</span> |data1={{M|\mathbb{P}[X\eq k]:\eq{}^n\text{C}_k\ p^k(1-p)^{n-k} }}
  4 KB (653 words) - 13:11, 22 September 2017
• The sum of two random variables with Poisson distributions is a Poisson distribution itself
  # {{M|X\sim\text{Poi}(\lambda)}} and # {{M|Y\sim\text{Poi}(r)}}
  3 KB (536 words) - 22:46, 4 November 2017
• Poisson race distribution
  ...X\sim}}[[Poisson distribution|{{M|\text{Poi} }}]]{{M|(\lambda_1)}} and {{M|Y\sim\text{Poi}(\lambda_2)}} be given {{plural|random variable|s}} (that are * {{M|Z:\eq X-Y}}
  6 KB (1,141 words) - 10:33, 24 December 2018
• Floor function
  For {{M|x\in\mathbb{R}_{\ge 0} }} there is no variation on the meaning of the floor f ...} }}]]{{M|(T_x)}} where {{M|T_x:\eq\big\{n\in\mathbb{N}_0\ \big\vert\ n\le x\big\}\subseteq\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0} }} - note that the ma
  2 KB (377 words) - 21:20, 21 January 2018
• Notes:Probability of an RV being less than another
  * {{MM|\P{X\le Y\le Z}\eq\sum_z\sum_{y\le z}\sum_{x\le y}\P{X\eq x\cap Y\eq y\cap Z\eq z} }} - ''duh!'' - silly me! # {{M|\P{X\le Y} }}, and
  2 KB (477 words) - 03:44, 12 December 2017
• Notes:Distribution of the sample median
  Let {{M|X_1,\ldots,X_{2m+1} }} be a sample from a population {{M|X}}, meaning that the {{M|X_i}} are {{iid}} [[random variables]], for some {{ Let us look at {{M|X\le r}} and {{M|X\le Y}} to see what we can say if both are true (the "{{link|and|logic}}")
  11 KB (2,371 words) - 17:21, 17 December 2017
• Projection matrix
  ...is to be [[transpose (matrix)|transposed]] to {{M|\left(\begin{array}{c}x\\y\end{array}\right)}} ...t {{M|v'\in\mathbb{R}^3}} be given, so {{M|v'\eq(x,y,z)^T}} for some {{M|x,y,z\in\mathbb{R} }}, now
  6 KB (1,051 words) - 07:44, 13 December 2017
• Identity map
  ...\text{Id}:X\rightarrow X}} is a [[function]] / [[map]] on some [[set]] {{M|X}}, then: * {{M|\forall x\in X[\text{Id}(x)\eq x]}}
  1 KB (235 words) - 15:06, 15 December 2017
• Arity
  If we write {{M|f(x,y)}} then {{M|f}} is a 2-ary function.
  401 B (59 words) - 15:10, 15 December 2017
• Notes:Borv
  ...s, then we define a [[random variable]] (with {{M|p}} as a parameter), {{M|X}}, as follows: * {{M|X\sim\text{Borv}(p)}}
  846 B (144 words) - 20:14, 1 January 2018
• Notes:Reflection of ray given normal
  * {{M|\ell_1}} by {{M|y:\eq mx+c}} and * {{M|\ell_2}} by {{M|y:\eq m'x+c'}}
  2 KB (444 words) - 21:02, 2 January 2018
• Poisson race distribution/CalcTable1
  ! {{M|X}} ! {{M|Y}}
  973 B (184 words) - 00:31, 7 January 2018
• Notes:Grid iteration
  ** {{M|x}}-coordinate: {{M|\text{Mix}_x(k,\ell,m\times n) }} ** {{M|y}}-coordinate: {{M|\text{Mix}_x\big(\text{Mix}_y(k,\ell,m\times n),m,n\big)}
  3 KB (468 words) - 19:23, 7 January 2018
• Rotation matrix
  ...orrespond to a rotation about one of the [[principle axies]], {{M|x}}, {{M|y}} or {{M|z}}, or a general vector.
  842 B (129 words) - 07:54, 3 April 2018
• Bastard's object
  * {{M|\forall X[\varphi(X)]}}^{In words:}<ref group="Note" name="FOLinWords1"/> made for some ...this means that {{M|\varphi(X)}} ''cannot'' be true '''''for all''''' {{M|X}}
  3 KB (569 words) - 22:15, 8 May 2018
• Kay (unit)
  ...and [[independent event (probability)|independently]] an event {{M|\mathrm{Y} }} occurs with rarity {{M|v}} ''kays'' then: * The event of both {{M|\mathrm{X} }} and {{M|\mathrm{Y} }} occurring has rarity {{M|u+v}}
  4 KB (573 words) - 19:42, 14 June 2018
• Poisson mixture
  ...} and {{M|\forall i\in\mathbb{N}\big[D_i\sim\text{Borv}(p)\big]}} then {{M|Y\sim\text{Poi}(p\lambda)}} is the distribution of the {{M|(D_i)_{i\in\mathbb * Define {{M|X}} to be an {{M|\mathbb{N} }}-valued random variable as follows:
  3 KB (509 words) - 00:41, 20 July 2018
• Minimum function
  ...eq\text{min}(u,v)\iff\big(x\preceq u \wedge x\preceq v \wedge (x\eq u \vee x\eq v)\big)\big]}} *** Unproven lemma: {{MM|\big(x\preceq u\wedge x\preceq v)\implies x\preceq \text{min}(u,v)}} - via [[contrapositive]]
  3 KB (473 words) - 07:53, 28 July 2018
• Notes:Collision
  ...2}v_1'+\frac{1}{m_2}(m_1v_1+m_2v_2)}} (written in the form {{M|y\eq m\cdot x+c}}) As this is the equation for a line (where {{M|x\eq v_1'}} and {{M|y\eq v_2'}}) we can see for huge {{|v_1'}} we get huge {{M|v_2'}} values (but
  3 KB (666 words) - 11:19, 25 September 2018
• Binary multiplexor
  | {{M|x}} | {{M|y}}
  806 B (143 words) - 15:29, 29 October 2018
• Hexadecimal numbers
  ! {{M|X+Y}} ! colspan="16" | {{M|Y}}
  3 KB (389 words) - 18:38, 23 February 2019