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• Books:Notes On Set Theory - Second Edition - Yiannis Moschovakis
  267 B (44 words) - 00:05, 4 February 2017

Page text matches

• Injection
  ...rjection/injection/[[bijection]] to be seen through the lens of [[Category Theory]]. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:50, 8 May 2018 (UTC) ==Notes==
  3 KB (463 words) - 21:50, 8 May 2018
• Basis for a topology
  ...but "let {{M|A\in\mathcal{P}(B)}}" instead. To emphasise that the [[power-set]] is possibly in play. ...se]], we usually deal with subsets of the ''space'' not subsets of the ''[[set system]]'' on that space.<br/>
  5 KB (802 words) - 18:35, 17 December 2016
• Relation
  ** For example {{M|<}} is a relation in the set of {{M|\mathbb{Z} }} (the integers) ! Set relation
  4 KB (762 words) - 20:07, 20 April 2016
• Equivalence relation
  * An [[equivalence class]] is the name given to the set of all things which are equivalent under a given equivalence relation. **[[The equivalence classes of an equivalence relation partitions a set]].
  3 KB (522 words) - 15:18, 12 February 2019
• Index of notation
  ! Notes ! [[Index of set-like notations]]
  9 KB (1,490 words) - 06:13, 1 January 2017
• Sequence
  ...>\{a_n\}_{n=1}^\infty</math> however I don't like this, as it looks like a set. I have seen the notation <math>(a_n)_{n=1}^\infty</math> and I must say I ...Maurin</ref>, <math>f:\mathbb{N}\rightarrow S</math> where {{M|S}} is some set. For a finite sequence it is simply <math>f:\{1,...,n\}\rightarrow S</math>
  2 KB (419 words) - 18:12, 13 March 2016
• Group
  A group is a set {{M|G}} and an operation <math>*:G\times G\rightarrow G</math>, denoted <ma If the operation is obvious then "Let {{M|G}} be the set of (whatever) and let {{M|(G,+)}} be a group"
  7 KB (1,332 words) - 07:17, 16 October 2016
• Algebra of sets
  * [[Types of set algebras]] ==Notes==
  3 KB (507 words) - 18:43, 1 April 2016
• Sigma-algebra
  # Show a {{sigma|algebra}} is closed under [[set-subtraction]], {{M|\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}} * {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]]
  8 KB (1,306 words) - 01:49, 19 March 2016
• Additive function
  {{Requires references|See Halmos' measure theory book too}} ...ve function (which way have meaning in say algebra), be sure to update the SET FUNCTION redirects that point into this page
  6 KB (971 words) - 18:16, 20 March 2016
• Measure
  A (positive) ''measure'', {{M|\mu}} is a [[set function]] from a [[sigma-ring|{{sigma|ring}}]], {{M|\mathcal{R} }}, to the ...n\right)=\sum_{n=1}^\infty\mu(A_n)]}} ({{M|\mu}} is a [[countably additive set function]])
  6 KB (941 words) - 14:39, 16 August 2016
• Measurable space
  {{Refactor notice|grade=A*|msg=Lets get this measure theory stuff sorted. At least the skeleton Given a [[set]], {{M|X}}, and a [[sigma-algebra|{{sigma|algebra}}]], {{M|\mathcal{A}\in\m
  2 KB (248 words) - 13:05, 2 February 2017
• Semi-ring of sets/Definition
  A collection of {{plural|set|s}}, {{M|\mathcal{F} }}<ref group="Note">An F is a bit like an R with an un ...mS_i}}. We require that they be pairwise disjoint AND their union be the [[set difference]] of {{M|S}} and {{M|T}}.</ref> - this doesn't require {{M|S-T\i
  2 KB (337 words) - 17:25, 18 August 2016
• The fundamental group
  {{Definition|Topology|Homotopy Theory}} * <math>\pi_1(X,x_0)</math> denotes the set of [[Homotopy class|homotopy classes]] of [[Paths and loops in a topologica
  3 KB (393 words) - 16:10, 4 November 2016
• Passing to the quotient (function)
  Suppose that {{M|1=w^{-1}(v)}} is [[empty-set|empty]] or contains 2 (or more!) elements, then what do we define {{M|\tild * To see that '''if {{M|f}} is surjective so is {{M|\tilde{f} }} see my notes here:
  8 KB (1,644 words) - 20:49, 11 October 2016
• Types of set algebras
  ==Measure theory perspective== ...e class of sets {{M|\mathcal{A} }} is a collection of subsets from another set {{M|\Omega}}
  3 KB (449 words) - 20:06, 19 August 2016
• Set
  * See [[Notes:Set]] {{Definition|Set Theory}}
  60 B (9 words) - 22:28, 19 August 2016
• Semi-ring of half-closed-half-open intervals
  ...]] of [[set|sets]] where one or more of the {{M|X_\alpha}} are the [[empty set]], {{M|\emptyset}}, then: ==Notes==
  4 KB (680 words) - 00:23, 20 August 2016
• Index of properties
  | {{M|\backslash}}-closed<ref name="PTACC">Probability Theory - A comprehensive course - Second Edition - Achim Klenke</ref> ...{{M|\backslash}}-closed uses {{M|\backslash}} to denote [[Set subtraction|set subtraction]]<ref group="Note">This is because {{M|-}}-closed is not a good
  2 KB (360 words) - 20:43, 15 June 2015
• Class of sets closed under complements properties
  ...a system of subsets of {{M|\Omega}} such that<ref name="PTACC">Probability Theory - A comprehensive course - Achim Klenke</ref>: * [[Types of set algebras]]
  1 KB (165 words) - 20:50, 15 June 2015
• Properties of classes of sets closed under set-subtraction
  Suppose {{M|\mathcal{A} }} is an arbitrary class of [[set|sets]] with the property that: ...=\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}} where {{M|A-B}} denotes "[[set subtraction]]" ({{AKA}}: [[relative complement]])
  3 KB (490 words) - 11:38, 21 August 2016
• Sigma-algebra generated by
  See [[Notes:Just what is in a generated sigma-algebra|Just what is in a generated {{Sig ...set|power set]] of {{M|\Omega}}) there exists<ref name="PTACC">Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke</ref> a smallest [
  2 KB (286 words) - 22:02, 17 March 2016
• Partition (abstract algebra)
  ...ry [[Indexing set|indexing set]], to each element {{M|i\in I}} we assign a set {{M|A_i}} which is non-empty. If: ...Analysis - Part 1: Elements - Krzystof Maurin</ref> ''the partition of the set {{M|B}} into classes {{M|A_i}} for {{M|i\in I}}''
  2 KB (316 words) - 16:03, 18 June 2015
• Group action
  A (left) ''group action'' of a [[group]] {{M|(G,*)}} on a [[set]] {{M|X}} is a [[mapping]]{{rAAPAG}}: ** [[The symmetric group on a set acts on the set by evaluation]]
  2 KB (320 words) - 23:28, 21 July 2016
• Dynkin system generated by
  Given a set {{M|X}} and another set {{M|\mathcal{G}\subseteq\mathcal{P}(X)}} which we shall call the ''generato {{Todo|Be bothered, notes on p75 of my notebook, or page 31 of<ref name="MIM"/>}}
  2 KB (245 words) - 15:16, 16 December 2016
• Well-ordered set
  A set {{M|A}} with an [[Linear ordering|linear ordering]] {{M|<\subseteq A\times ==Notes==
  916 B (164 words) - 17:49, 24 July 2015
• Dynkin system
  * [[Types of set algebras]] ==Notes==
  1 KB (184 words) - 01:54, 19 March 2016
• Dynkin system/Definition 2
  <noinclude>{{Extra Maths}}</noinclude>Given a set {{M|X}} and a family of subsets of {{M|X}} we denote {{M|\mathcal{D}\subset ==Notes==
  719 B (120 words) - 23:22, 2 August 2015
• Function terminology
  ...ion {{M|f:A\rightarrow B}} where {{M|A}} and {{M|B}} are assumed to be any set unless stated otherwise we say the following about {{M|f}}: ==Notes==
  1 KB (165 words) - 02:07, 3 August 2015
• Borel sigma-algebra generated by
  ...used, however in measure theory this notation is often used to denote the set of half-open-half-closed rectangles in {{M|\mathbb{R}^n}} - a totally separ ==Notes==
  2 KB (244 words) - 08:30, 6 August 2015
• Borel sigma-algebra
  ...illing</ref>. It is [[Sigma-algebra generated by|generated by]] the [[Open set|open sets]] of the [[Metric space|metric space]] {{M|(\mathbb{R},\vert\cdot ...">Conventionally, {{M|\mathcal{J} }} denotes the open sets, but in measure theory this seems to denote the sets of half-open-half-closed rectangles, and it i
  5 KB (854 words) - 09:25, 6 August 2015
• The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set
  ...than the sum of the (pre-)measures of the elements of a covering for that set/Statement|Statement]]== ...than the sum of the (pre-)measures of the elements of a covering for that set/Statement}}
  4 KB (688 words) - 21:03, 31 July 2016
• Category
  # For every ordered pair, {{M|(X,Y)}} of ''objects'' a set {{M|\hom(X,Y)}} of ''morphisms'' {{M|f}} ==Notes==
  2 KB (347 words) - 00:36, 27 September 2015
• Classes of continuously differentiable functions
  Given {{M|U\subseteq\mathbb{R}^n}} (where {{M|U}} is [[Open set|open]]) and some {{M|k\ge 0}}, a function of the form: ...''"...functions of class {{M|C^k}} on {{M|U}} by..."'' when describing the set {{M|C^k(U)}}</ref> we require<ref name="ITSM"/>:
  3 KB (632 words) - 20:32, 16 October 2015
• Pre-measure/New page
  ...ive) pre-measure'' is an ''[[extended real valued]]'' [[countably additive set function]], {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0 * [[Types of set algebras]]
  3 KB (422 words) - 21:25, 17 August 2016
• Subsequence/Definition
  {{Requires cleanup|grade=A|msg=The notes are a bit messy}} ==Notes==
  3 KB (484 words) - 21:54, 16 November 2016
• Cartesian product
  Given two sets, {{M|X}} and {{M|Y}} their ''Cartesian product'' is the set: ===Set construction===
  2 KB (318 words) - 14:25, 19 February 2016
• Strict partial ordering
  ...for anything other than denoting [[subset|subsets]], the relation and the set it relates on will go together, so you'll already be using {{M|\subseteq}} ==Notes==
  3 KB (436 words) - 10:15, 20 February 2016
• Preorder
  A ''preorder'', {{M|\preceq}}, on a set {{M|X}} is a [[relation]] in {{M|X}}, so {{M|\preceq\subseteq X\times X}}, A tuple, consisting of a set {{M|X}}, equipped with a preorder {{M|\preceq}} is called a ''[[preset]]''<
  2 KB (355 words) - 10:13, 20 February 2016
• Index of operators on ordered sets
  If you are given a set, say {{M|X}} and any of a: on that set, then this page indexes various operators that might take such a structured
  2 KB (304 words) - 17:01, 20 February 2016
• Predicate
  ...dicate'', {{M|P}}, is a [[n-place relation|{{M|1}}-place relation]] on a [[set]] {{M|X}}<ref group="Note">{{M|P\subseteq X}} in this case. In contrast to ...comprehension]] - This states that given a set {{M|A}} we can construct a set {{M|B}} such that {{M|1=B=\{x\in A\ \vert P(x)\} }} for some ''predicate''
  916 B (160 words) - 18:44, 18 March 2016
• Sigma-algebra/Definition
  </noinclude>Given a [[set]] {{M|X}} a ''{{sigma|algebra}} on {{M|X}}'' is a family of subsets of {{M| ==Notes==
  779 B (122 words) - 01:25, 19 March 2016
• Notes:Measures
  ...thors to see what is what. Bogachev for example (author of [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]) doesn't require that a measure even be posit # [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]
  5 KB (771 words) - 03:06, 21 March 2016
• Symmetric difference
  Let {{M|A,B\in\mathcal{P}(X)}} be two [[subset|subsets]] of a [[set]] {{M|X}}. We define the ''symmetric difference'' of {{M|A}} and {{M|B}} as ...A\triangle B:=(A-B)\cup(B-A)}}<ref group="Note">Here {{M|A-B}} denotes ''[[set subtraction]]''.</ref>
  830 B (139 words) - 00:59, 21 March 2016
• Notes:Halmos measure theory skeleton
  The "core" of Halmos' measure theory book is the following: # [[Measure]], {{M|\mu}}, countably additive extended real valued set function on a {{sigma|ring}}, {{M|\mathcal{R} }}, {{M|\mu:\mathcal{R}\right
  4 KB (581 words) - 20:31, 3 April 2016
• Notes:Measure theory plan
  This document is the ''plan'' for the measure theory notation and development on this site. ...mu} }}) - Introduce a (positive) extended real valued [[countably additive set function]], {{M|\bar{\mu} }} on that ring. This will be a pre-measure and t
  4 KB (619 words) - 19:28, 24 May 2016
• Site projects:Measure theory plan
  Inline with the [[Notes:Measure theory plan]] page we will use the terms and symbols which may be found on the bot * A page (perhaps [[measure theory and sigma algebras notice]] will need to (have part of it) be transcluded o
  2 KB (309 words) - 15:31, 27 March 2016
• Hereditary system of sets
  {{Stub page|Needs linking to where it is used, notes on a sort of "power-set" like construct.|grade=B}} ...in [[Measure Theory (subject)|Measure Theory]] and ''[[Hereditary (measure theory)]]'' redirects here
  793 B (125 words) - 21:25, 19 April 2016
• Restriction
  ...l{P}(X)}}<ref group="Note">Recall {{M|\mathcal{P}(X)}} denotes the [[power-set]] of {{M|X}}</ref> (so {{M|A\subseteq X}} - and is any subset) we define a ==Notes==
  652 B (107 words) - 20:01, 8 April 2016
• Extending pre-measures to outer-measures
  ...tly being written and is not ready for being used as a reference, it's a ''notes quality'' page}} ...infty A_n\right\} }} - here {{M|\text{inf} }} denotes the [[infimum]] of a set.
  11 KB (1,921 words) - 16:59, 17 August 2016
• Site projects:Split set theory into elementary set theory
  ...this way pages tagged as elementary set theory have no business in the set theory category. * Create an "elementary set theory" subject in the [[Site:Mathematical subject index]]
  739 B (114 words) - 12:21, 9 April 2016
• Infimum
  ...athcal{P}(X)}} where {{M|\mathcal{P}(S)}} denotes the [[power set]] of a [[set]] {{M|S}}</ref>. The ''infimum'' ({{AKA}}: ''greatest lower bound'', ''g.l. ...ce{\left\{x\in X\ \vert\ (\forall a\in A[x\preceq a])\right\} }_{\text{the set of all lower bounds of }A }\Big[b\preceq\text{Inf}(A)\Big]}} - which states
  5 KB (851 words) - 08:55, 29 July 2016
• Notes:First order language
  ...eference names used on this page are not the usual ones, as this is just a notes page ## Equality symbol - the set containing {{M|\doteq}}<ref name="WL"/><ref group="Note">This is done so th
  6 KB (1,088 words) - 09:22, 28 August 2016
• Notes:Outer-measures to measures
  * {{M|\bar{S} }} - the set of all outer-measurable sets, WRT {{M|\mu^*}} (for the definition of outer- ...f he defines {{M|E_n}} as being a term "{{M|E_n}}" that happens to be in a set.... it's iffy at best (then {{M|E_j}} wouldn't make sense)
  6 KB (1,067 words) - 22:19, 23 May 2016
• Hereditary sigma-ring generated by
  ...|grade=A*|msg=Needed for progress, I started the page to get some notation set in stone.}} * {{M|1=\mathcal{H}(\sigma_R(S))=\sigma_R(\mathcal{H}(S))}} - see [[Notes:Hereditary sigma-ring]]
  877 B (138 words) - 19:24, 24 May 2016
• The set of all mu*-measurable sets is a sigma-ring
  {{Stub page|grade=A*|msg=Currently in the notes stage, see [[Notes:The set of all mu*-measurable sets is a ring]]}} ...0} }} (where {{M|\mathcal{H} }} is a [[hereditary sigma-ring]]) that [[the set of all mu*-measurable sets is a ring]]. It is in fact not only a [[ring of
  521 B (82 words) - 01:01, 30 May 2016
• Notes:The set of all mu*-measurable sets is a ring
  In [[Books:Measure Theory - Paul R. Halmos|Measure Theory]] Halmos does something weird for section 11, theorem B. I have yet to "cra ...mathcal{H} }} and if {{M|\mathcal{S} }} is the set of all [[mu*-measurable set|{{M|\mu^*}}-measurable sets]] then {{M|\mathcal{S} }} is a [[sigma-ring]].
  4 KB (828 words) - 03:11, 30 May 2016
• Notes:Measures - Real and Abstract Analysis - Hewitt & Stromberg
  A continuation of the [[Notes:Measures]] series, covering the terminology used in [[Books:Real and Abstra ...{M|X}}, a function, {{M|\mu}} defined on {{M|\mathcal{P}(X)}} (the [[power-set]] of {{M|X}}) is a Carathéodory Outer Measure if:
  452 B (71 words) - 18:04, 31 May 2016
• Group/New page
  Let {{M|G}} be a [[set]] and a [[binary operation]] (a [[function]]) {{M|*:G\times G\rightarrow G} ==Notes==
  2 KB (326 words) - 11:38, 2 July 2016
• Notes:Equivalence relations
  This is just some notes to get what I've done on paper into the system 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
  ...play are eligible (satisfy the requirements to factor) for the theorem. We set up as follows: ...epresent the classes in what follows, however as has been mentioned in the notes several times, this isn't a problem.
  7 KB (1,195 words) - 22:55, 3 December 2016
• Free monoid generated by
  ...to check [[Discussion of the free monoid and free semigroup generated by a set]], as there are some things to note Given a [[set]], {{M|X}}, there is a ''free'' [[monoid]], {{M|(F,*)}}{{rAAPAG}}.
  2 KB (419 words) - 16:20, 20 July 2016
• Notes:Advanced Linear Algebra - Roman/Chapter 1
  ==Notes== * The set of all finite sums of vectors from the [[union]] {{MM|1=\bigcup_{\alpha\in
  8 KB (1,463 words) - 14:35, 13 August 2016
• Extending pre-measures to measures
  # [[The set of all mu*-measurable sets forms a ring|the set of all {{M|\mu^*}}-measurable sets forms a ring]] # [[The set of all mu*-measurable sets forms a sigma-ring|the set of all {{M|\mu^*}}-measurable sets forms a {{sigma|ring}}]]
  2 KB (257 words) - 17:27, 17 August 2016
• The ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions
  ...tion</ref> {{M|\mathcal{F} }}, written {{M|R(\mathcal{F})}} is exactly the set {{M|\mathcal{F} }} and all finite [[union|unions]] of elements of {{M|\math ...e proof of this is easy, as [[the intersection of sets is a subset of each set]] we see {{M|1=A\cap B_i\subseteq B_i}} for each {{M|i}}. As the {{M|B_i}}
  7 KB (1,398 words) - 18:33, 19 August 2016
• A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
  * The ring generated by a semi-ring is exactly the set of all finite disjoint unions of elements from that semiring. # [[the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions]]
  2 KB (390 words) - 22:16, 19 August 2016
• Types of set algebras/Type table
  ...e, if you have come here from a search engine you want to go to [[types of set algebras]]''' ! [[Ring of sets|Ring]]<ref name="PTACC">Probability Theory - A comprehensive course - second edition - Achim Klenke</ref>{{rMTH}}
  4 KB (573 words) - 20:00, 19 August 2016
• Doctrine:Measure theory terminology/Proposals
  This is a sub page for making proposals to the measure theory terminology doctrine. New requests only must be placed here. Queries and su I propose that rather than {{plural|mu*-measurable set|s}} we instead use [[outer splicing sets]] or just [[splicing sets]]. Curre
  3 KB (466 words) - 21:29, 20 August 2016
• Outer splicing set
  # Unite this with the [[mu*-measurable set]] page, possibly by redirecting it here ...t. It is not a well known term. [[mu*-measurable set|{{M|\mu*}}-measurable set]] redirects here.
  2 KB (378 words) - 22:09, 20 August 2016
• Partial-function
  ...">This is my own term. With total orderings any two elements of underlying set of the relation must be comparable. With a total function, {{M|g}}, {{M|g}} ...} (here {{M|f^{-1}(B)}} denotes the [[pre-image]] of {{M|B}}, which is the set containing all {{M|a\in A}} such that {{M|f}} relates {{M|a}} to a {{M|b\in
  2 KB (462 words) - 22:26, 23 August 2016
• Hereditary system generated by
  : {{Warning|This page is little more than notes at the moment, however everything stated here is verified and correct}} ...\in\mathcal{P}(B))]}}</ref> where {{M|\mathcal{P}(A)}} denotes the [[power set]] of {{M|A}}.
  2 KB (371 words) - 23:36, 23 August 2016
• First order language
  {{Provisional page|grade=A|msg=Needed for set theory}} : There are [[Notes:First order language|notes on this topic]]
  3 KB (455 words) - 10:45, 8 September 2016
• Domain (FOL)
  ...ty bad that this requires a notion of sets when I want to use this for set theory}} * {{M|M}} is a ''[[non-empty]]'' [[set]] {{Caution|I am studying this for set theory, so something is needed here}}
  4 KB (672 words) - 06:42, 10 September 2016
• Demonstrating why category arrows are best thought of as arrows and not functions
  ...ry|s}} are ''[[finite]]'' {{plural|set|s}} and whose {{link|arrow|category theory|s}}, {{M|\xymatrix{A \ar[r]^f & B} }} are {{plural|function|s}}{{rAITCTHS20 ==Notes==
  2 KB (275 words) - 12:29, 15 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
  ** Let {{M|C^0(X,Y)}} denote the [[set]] of all [[continuous maps]] of the form {{M|(:X\rightarrow Y)}} ==Notes==
  2 KB (272 words) - 23:37, 14 October 2016
• Doctrine:Homotopy terminology
  * Let {{M|C^0(X,Y)}} denote [[the set of continuous maps between spaces|the set of continuous maps between {{Top.|X|J}} and {{Top.|Y|K}}]]<ref group="Note" ==Notes==
  3 KB (535 words) - 09:01, 31 October 2016
• Every injection yields a bijection onto its image
  ==Notes== * {{M|1=f(X):=\{y\in Y\ \vert \exists x\in X[f(x)=y] \} }} - the set of all things in {{M|Y}} that are mapped to by {{M|f}} for some {{M|x\in X}
  4 KB (813 words) - 11:53, 26 September 2016
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
  # We must check the set up satisfies the requirements of the {{link|passing-to-the-quotient|functio ==Notes==
  6 KB (1,097 words) - 20:24, 9 October 2016
• Exercises:Mond - Topology - 1/Question 8
  Which theorem of [[Group Theory (subject)|group theory]] does this resemble? ...]], then {{M|\frac{X}{\sim} }} is compact also as [[the image of a compact set is compact]] and {{M|\pi:X\rightarrow\frac{X}{\sim} }} is continuous (see [
  3 KB (413 words) - 00:13, 12 October 2016
• Ring/New page
  Let {{M|R}} be a ''[[non-empty]]'' [[set]], let there be two [[binary operations]] (a kind of [[map]] where rather t ==Notes==
  4 KB (728 words) - 16:29, 19 October 2016
• Equivalent conditions to a set being saturated with respect to a function
  ==Notes== {{Theorem Of|Elementary Set Theory|Set Theory}}
  807 B (142 words) - 13:15, 16 October 2016
• Direct product module
  * {{M|\prod_{\alpha\in I}M_\alpha}} is the underlying set of the module (we define {{M|1=M:=\prod_{\alpha\in I}M_\alpha}} for conveni * [[Direct sum of modules]] - instances of a {{link|co-product|category theory}}
  3 KB (431 words) - 22:19, 19 October 2016
• Constant loop based at a point
  #* Let {{M|U\in\mathcal{J} }} be given (so {{M|U}} is an [[open set]] in {{Top.|X|J}}) ==Notes==
  3 KB (479 words) - 21:03, 1 November 2016
• Constant function
  ==Notes== {{Definition|Elementary Set Theory|Set Theory}}[[Category:Function Terminology]]
  977 B (169 words) - 21:12, 1 November 2016
• Proof that the fundamental group is actually a group
  ...{M|b\in X}} be given. Then [[Omega(X,b)|{{M|\Omega(X,b)}}]] denotes the "''set of all [[loops]] based at {{M|b}} in {{M|X}}''"<ref group="Note">Which is a ==Notes==
  2 KB (281 words) - 11:20, 8 November 2016
• Proof that the fundamental group is actually a group/Outline
  ...note the relation of [[end-point-preserving homotopy]] on {{C(I,X)}} - the set of all {{link|path|topology|s}} in {{M|X}} - but considered only on the sub ==Notes==
  3 KB (454 words) - 18:31, 4 November 2016
• Homotopy invariance of loop concatenation
  ...ll(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 on it, called [[the fund ==Notes==
  3 KB (462 words) - 09:21, 6 November 2016
• Invariant of an equivalence relation
  ...bra]] where {{M|W}} is quite often used for [[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from {{M|S}} to {{M|W}}. * [[Complete system of invariants]] - a finite set of complete invariants really.
  3 KB (478 words) - 18:58, 9 November 2016
• Quotient by an equivalence relation
  Let {{M|X}} be a [[set]] and let {{M|\sim\subseteq X\times X}} be an [[equivalence relation]] on { Yes, {{M|\frac{X}{\sim} }} is the set of equivalence classes, it is that simple.
  2 KB (295 words) - 14:16, 13 November 2016
• Equivalence classes are either equal or disjoint
  ...{{M|\sim}}]]<ref group="Note">In other words: {{M|\frac{X}{\sim} }} is the set of [[equivalence classes]] of {{M|\sim}}</ref> and lastly let {{M|\pi:X\rig ==Notes==
  2 KB (388 words) - 14:36, 13 November 2016
• Polynomial u-ring
  ...be a [[u-ring]] with the standard operations of {{M|+}} and {{M|*}}. The ''set (or further-more: u-ring) of polynomials over {{M|R}}'' (in the ''indetermi ...bject is countable]] and [[finitely many elements removed from a countable set remains countable]]
  3 KB (643 words) - 02:34, 20 November 2016
• A continuous map induces a homomorphism on fundamental groups
  ...{{M|\mathbb{P}_X^{-1}([f]) }} in the notation developed next, giving us a set of all things equivalent to {{M|f}} and for any of these {{M|\varphi_\ast}} ...ef><ref group="Note">Recall that [[Omega(X,b)|{{M|\Omega(X,p)}}]] is the [[set]] of all {{link|loop|topology|s}} in {{M|X}} based at {{M|p\in X}}. There i
  8 KB (1,475 words) - 07:35, 14 December 2016
• Order terminology
  If {{M|(X,\preceq)}} is a [[poset]], a set with a [[partial order]], then we get a [[strict partial order]], {{M|\prec ...t a total order provided you are looking at sets in the [[power set]] of a set with more than one element). Notice that for subsets of {{M|\{1,2,3\} }} th
  4 KB (656 words) - 18:45, 8 January 2017
• Notes:Delta complex/Formal attempt
  * Let {{M|S_n(K)}} be the set of {{n|{{plural|simpl|ex|ices}}}} of {{M|K}} ...ork including this and come up with 2 "null objects" that do not alter the theory, for now {{M|\text{Dim}(\emptyset)\eq -1}} will do. It wont matter.
  5 KB (966 words) - 14:36, 6 February 2017
• Abstract simplicial complex
  ...t time as it will not come up in my exam! However I think it is important! Notes for improvement: ...in\mathbb{N}_0]}} - where {{M|\vert\cdot\vert}} denotes [[cardinality of a set]]
  3 KB (428 words) - 11:54, 19 February 2017
• Vertex scheme of an abstract simplicial complex/Definition
  ...set|simplicial complex}} of {{M|K}} (not to be confused with the [[vertex set of an abstract simplicial complex]]), then we may define {{M|\mathcal{K} }} ...needs to be rewritten!}}</ref> - that is to say {{M|\mathcal{K} }} is the set containing all collections of vertices such that the vertices span a simple
  1 KB (224 words) - 11:51, 19 February 2017
• Evenly covered by a continuous map
  ...inuous map]]. Let {{M|U\in\mathcal{J} }} be given, so {{M|U}} is an [[open set]] of {{Top.|X|J}}, we say that: * Notice that {{M|f^{-1}(U)}} is [[open set|open]] as {{M|f}} is continuous, and that the union of an arbitrary family
  2 KB (394 words) - 21:54, 24 February 2017
• Unique lifting property
  ...cted neighbourhoods. It's worth investigating but it isn't critical to the theory.</ref> ...e see that the ''only'' sets that are both [[open sets|open]] and [[closed set|closed]] are {{M|Y}} itself and {{M|\emptyset}}, if the result holds (which
  13 KB (2,510 words) - 16:23, 2 March 2017
• Simple function under-approximation to a numerical function
  : '''See notes page: ''' [[Notes:Simple function approximation to a numerical function]] {{Stub page|grade=A|msg=Important for measure theory, and needs a name. SNAF {{M|\leftarrow}} ''simple numerical approximation f
  2 KB (289 words) - 15:28, 3 March 2017
• Singleton (set theory)/Definition
  ...">[[Media:WarwickSetTheoryLectureNotes2011.pdf|Warwick lecture notes - Set Theory - 2011 - Adam Epstein]] - page 2.75.</ref>: ...X[t\eq s]}}<ref group="Note">see [[rewriting for-all and exists within set theory]]</ref>
  1 KB (219 words) - 23:34, 8 March 2017