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Page title matches

• Topological space
  ==[[Topological space/Definition|Definition]]== {{:Topological space/Definition}}
  2 KB (268 words) - 13:37, 20 April 2016
• Metric space
  A [[Normed space|normed space]] is a special case of a metric space, to see the relationships between metric spaces and others see: [[Subtypes ==Definition of a metric space==
  2 KB (336 words) - 06:07, 27 November 2015
• Dual vector space
  ...tor space]] {{M|(V,F)}} we define the '''dual''' or '''conjugate''' vector space<ref name="LAVEP">Linear Algebra via Exterior Products - Sergei Winitzki</re Theorem: Given a basis {{M|1=\{e_1,\cdots,e_n\} }} of a vector space {{M|(V,F)}} there is a corresponding basis to {{M|V^*}}, {{M|1=\{e_1^*,\cdo
  3 KB (614 words) - 05:35, 8 December 2016
• Vector space
  A vector space {{M|V}} over a [[Field|field]] {{M|F}} is a non empty set {{M|V}} and the b Such that the following 8 "axioms of a vector space" hold
  2 KB (421 words) - 16:30, 23 August 2015
• The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
  ...be a [[metric]] on that set and let {{M|(X,d)}} be the resulting [[metric space]]. Then we claim: {{Theorem Of|Topology|Metric Space}}
  4 KB (814 words) - 22:16, 16 January 2017
• Measure space
  '''Note:''' This page requires knowledge of [[Measurable space|measurable spaces]]. A ''measure space''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</r
  1 KB (188 words) - 15:24, 21 July 2015
• Measurable space
  * {{M|A\subseteq\mathcal{P}(X)}}</ref> then a ''measurable space''{{rMIAMRLS}}{{rAGTARAF}} is the [[tuple]]: ...(X,\mathcal{A},\mu)}} where {{M|\mu}} is a [[measure]] on the ''measurable space'' {{M|(X,\mathcal{A})}}
  2 KB (248 words) - 13:05, 2 February 2017
• Probability space
  Given a [[Measure space|measure space]] {{M|(X,\mathcal{A},\mu)}} where <math>\mu</math> is a [[Probability measu Now {{M|(X,\mathcal{A},\mathbb{P})}} is a ''Probability space''
  2 KB (338 words) - 22:55, 2 May 2015
• Tangent space
  ==Types of tangent space== | [[Tangent space#Geometric Tangent Space|Geometric tangent space]]
  6 KB (1,190 words) - 19:27, 14 April 2015
• Motivation for tangent space
  {{Todo| get back to tangent space}}
  6 KB (975 words) - 00:18, 11 April 2015
• Motivation for tangent space definitions
  '''Note: ''' different to [[Motivation for tangent space]] - that page talks about tangents, and going between manifolds. THIS page ==Why have geometric tangent space?==
  4 KB (790 words) - 22:25, 12 April 2015
• Covering space
  #REDIRECT [[Topological covering space]] Here {{M|(E,\mathcal{K})}} and {{M|(X,\mathcal{J})}} are [[Topological space|topological spaces]]
  1 KB (213 words) - 01:26, 26 February 2017
• Paths and loops in a topological space
  ==Path in a topological space== ===Loop in a topological space===
  2 KB (347 words) - 19:36, 16 April 2015
• Hausdorff space
  Given a [[topological space]] {{M|(X,\mathcal{J})}} we say it is ''Hausdorff''{{rITTBM}} or ''satisfies * It may also be said that in a Hausdorff space that "''points may be separated by open sets''"{{rITTMJML}}
  4 KB (679 words) - 22:52, 22 February 2017
• Hilbert space
  ...th>\langle\cdot,\cdot\rangle</math> such that {{M|H}} is [[Complete metric space|complete]] with respect to the associated [[Norm|norm]] <math>\|x\|=\sqrt{\ That is to say a Hilbert space is a [[Banach space]] where the norm is given by an inner product
  573 B (93 words) - 17:34, 21 April 2015
• Notes:Vector space operations
  This page is a notes page for evidence of operations on [[Vector space|vector spaces]] - that is the definitions of these operations according to | Vector space on tuples of vectors
  3 KB (489 words) - 20:27, 1 June 2015
• Compact subspace of a Hausdorff space is closed
  23 B (4 words) - 04:52, 22 June 2015
• Closed set in a compact space is compact
  27 B (3 words) - 00:31, 25 June 2015
• Set in a compact Hausdorff space is compact iff it is closed
  43 B (4 words) - 00:31, 25 June 2015
• Compact set in a Hausdorff space is closed
  41 B (4 words) - 00:33, 25 June 2015

Page text matches

• Topology
  ...hen talk of topologies we don't mean a topology but rather a [[topological space]] which is a topology with its underlying set. See that page for more detai A [[topological space]] is simply a [[tuple]] consisting of a set (say {{M|X}}) and a topology (s
  3 KB (543 words) - 09:28, 30 December 2016
• Homeomorphism
  If {{Top.|X|J}} and {{Top.|Y|K}} are [[topological space|topological spaces]] a ''homeomorphism from {{M|X}} to {{M|Y}}'' is a{{rITT '''Claim 1:''' {{M|\cong}} is an [[equivalence relation]] on [[topological space|topological spaces]].
  5 KB (731 words) - 22:58, 22 February 2017
• Topological space
  ==[[Topological space/Definition|Definition]]== {{:Topological space/Definition}}
  2 KB (268 words) - 13:37, 20 April 2016
• Open set
  ===Topological space=== In a [[topological space]] {{M|(X,\mathcal{J})}} we have:
  4 KB (677 words) - 02:26, 29 November 2015
• Open ball
  Given a [[metric space]] {{M|(X,d)}} the ''open ball centred at {{M|x_0\in X}} of radius {{M|r>0}} ...ns denote an open ball of radius {{M|r}} centred at {{M|x}} (in a [[metric space]] {{M|(X,d)}}, this table is supposed to be complete, so preferred notation
  4 KB (842 words) - 02:00, 29 November 2015
• Metric space
  A [[Normed space|normed space]] is a special case of a metric space, to see the relationships between metric spaces and others see: [[Subtypes ==Definition of a metric space==
  2 KB (336 words) - 06:07, 27 November 2015
• Limit point
  {{Definition|Topology|Metric Space}} For a [[Topological space]] <math>(X,\mathcal{J})</math>, <math>x\in X</math> is a limit point of <ma
  877 B (133 words) - 14:09, 16 June 2015
• Continuous map
  Given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} we say Again, given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a
  6 KB (972 words) - 01:44, 14 October 2016
• Dual vector space
  ...tor space]] {{M|(V,F)}} we define the '''dual''' or '''conjugate''' vector space<ref name="LAVEP">Linear Algebra via Exterior Products - Sergei Winitzki</re Theorem: Given a basis {{M|1=\{e_1,\cdots,e_n\} }} of a vector space {{M|(V,F)}} there is a corresponding basis to {{M|V^*}}, {{M|1=\{e_1^*,\cdo
  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}}: '''Recall''' the definition of a topological space being ''{{link|disconnected|topology}}''
  5 KB (866 words) - 01:52, 1 October 2016
• Compactness
  # We may only say a [[topological space]] is compact, we may not speak of the compactness of subsets. Compactness i # Sure talk about the compactness of subsets of a space.
  5 KB (828 words) - 15:59, 1 December 2015
• Subspace topology
  ...equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\mathcal{J}_S)}} where the [[topology]] {{M|\mathcal{J}_S}} is know Given a [[Topological space|topological space]] {{M|(X,\mathcal{J})}} and given a {{M|Y\subset X}} ({{M|Y}} is a subset o
  6 KB (1,146 words) - 23:04, 25 September 2016
• Quotient topology
  If <math>(X,\mathcal{J})</math> is a [[Topological space|topological space]], <math>A</math> is a set, and <math>p:(X,\mathcal{J})\rightarrow A</math> Let {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} be [[Topological space|topological spaces]] and let {{M|p:X\rightarrow Y}} be a [[Surjection|surje
  5 KB (795 words) - 13:34, 16 October 2016
• Motivation for topology
  ...he definition of [[Continuous map|continuity]], on a [[Metric space|metric space]] <math>\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\imp ...gh. (this motivates the "union of open sets is open" part of [[Topological space|topologies]])
  1 KB (243 words) - 15:39, 13 February 2015
• Continuity definitions are equivalent
  ...ction <math>f:(X,d)\rightarrow(Y,d')</math> from one [[Metric space|metric space]] to another is the same as <math>f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K
  2 KB (476 words) - 07:20, 27 April 2015
• Basis for a topology
  Let {{Top.|X|J}} be a [[topological space]] and let {{M|\mathcal{B}\in\mathcal{P}(\mathcal{P}(X))}} be any collection ...ith subsets of the ''space'' not subsets of the ''[[set system]]'' on that space.<br/>
  5 KB (802 words) - 18:35, 17 December 2016
• Basis
  * [[Basis (vector space)]]
  229 B (31 words) - 13:40, 16 August 2016
• Closed set
  A closed set in a [[Topological space|topological space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X-A</mat ===Metric space===
  1 KB (238 words) - 15:36, 24 November 2015
• Covering
  {{Definition|Topology|Metric Space}}
  190 B (24 words) - 15:15, 2 December 2015
• Homomorphism
  ** [[Homomorphism (vector space)]] - {{AKA}}: [[linear map]] - instance of a [[module homomorphism]] ** {{plural|continuous map|s}} - the homomorphisms of {{plural|topological space|s}} (not to be confused with [[homeomorphism]]) - see also: [[TOP (category
  4 KB (532 words) - 22:04, 19 October 2016