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• 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
• Closure of a set in a topological space
  ...sent here and then turn that into an overview page. This page is marked A* in grade because of the importance of the closure, interior and boundary conce ..., boundary is the reason {{C|closure (topology)}} couldn't be used as even in [[Topology (subject)|topology]] "boundary" has several meanings.
  2 KB (256 words) - 10:16, 28 September 2016
• Disjoint in a set
  #REDIRECT [[Disjoint#Disjoint in a set]] {{Definition|Elementary Set Theory|Set Theory}}
  88 B (12 words) - 03:19, 1 October 2016
• A set is open if and only if every point in the set has an open neighbourhood contained within the set
  ...general [[neighbourhood]] one, if I could just be bothered to prove, "''a set is open if and only if it is neighbourhood to all of its points''" or somet Let {{Top.|X|J}} be a [[topological space]], then{{rITTMJML}}:
  675 B (114 words) - 06:06, 14 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
  ...rade=C|msg=Obvious and easy "theorem", created to make the proof of claims in [[Dense]] more applicable to other pages and thus worth covering}} Let {{M|A}} and {{M|B}} be [[sets]]. Then we claim:
  2 KB (287 words) - 19:31, 28 October 2016
• 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
  : This is a [[precursor theorem]] to "''[[a proper vector subspace of a topological vector space has no interior]]''". ...]{{M|)}} be a [[topological vector space]] and let {{M|(Y,\mathbb{K})}} be a [[vector subspace]] of {{M|(X,\mathbb{K})}}, then{{rFAVIDMH}}:
  924 B (140 words) - 17:51, 16 February 2017
• The interior of a set in a topological space is equal to the union of all interior points of that set
  The [[interior (topology)|{{M|\text{Int} }}]]{{M|(A)}} is equal to the set of all interior points.<ref>Alec's own work - see the [[interior (topology)
  550 B (77 words) - 20:15, 16 February 2017
• A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
  ...set being bounded]] which should be moved to [[Equivalent conditions to a set being metrically bounded]] [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 00: Let {{M|(X,d)}} be a [[metric space]] and let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset of]] {{M|X}}. Then we claim{{rFAVI
  6 KB (1,092 words) - 00:41, 19 March 2017

Page text matches

• Main Page
  ...t interface by far is the search box''' this project is intended to create a ''queriable resource'' for looking up things and learning from them (and th Some pages may be rather sparse, containing little more than a definition, but there is always some information even if it is minimal. Thi
  7 KB (999 words) - 16:51, 11 May 2020
• Topology
  {{Stub page|grade=A*|msg=Should be easy to flesh out, find some more references and demote to g ...but rather a [[topological space]] which is a topology with its underlying set. See that page for more details}}
  3 KB (543 words) - 09:28, 30 December 2016
• Power set
  ...as notation. This comes from the [[Cardinality|cardinality]] of the power set being <math>2^{|X|}</math>) ...haracteristic property of the power set is that <math>\forall U\subset X:U\in\mathcal{P}(X)</math>
  492 B (92 words) - 16:30, 23 August 2015
• Injection
  {{Requires work|grade=A* |msg=This needs to be modified (in tandem with [[Surjection]]) to:
  3 KB (463 words) - 21:50, 8 May 2018
• Topological space
  {{Stub page|grade=A|msg=Hasn't been updated since March 2015, in April 2016 it was updated to modern format and cleaned up}} ...hat {{M|\mathcal{J}_1\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_1[S\in\mathcal{J}_2]}}
  2 KB (268 words) - 13:37, 20 April 2016
• Open set
  In a [[topological space]] {{M|(X,\mathcal{J})}} we have: ...J} }} that {{M|S}} is an open set. {{M|\mathcal{J} }} is by definition the set of open sets of {{M|X}}
  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}}'', denoted {{M|B_r(x_0)}} (however many notations a ...{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 {{M|x_0}}
  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 ==Definition of a metric space==
  2 KB (336 words) - 06:07, 27 November 2015
• Limit point
  ...of <math>x</math> has a non-empty [[Intersection|intersection]] with <math>A</math> that contains some point other than <math>x</math> itself. <math>x</math> is a limit point of <math>A</math> if <math>x\in\text{Closure}(A-\{x\})</math> (you can read about [[Closure, interior and boundary#Closure|
  877 B (133 words) - 14:09, 16 June 2015
• Continuous map
  {{Refactor notice|grade=A}} ...are a few different conditions for continuity, there's also continuity at a point. This diagram is supposed to show how they relate to each other.
  6 KB (972 words) - 01:44, 14 October 2016
• Surjection
  {{Requires work|grade=A*|msg=See [[Injection]]'s requires-work box [https://wiki.unifiedmathematics ...element of <math>B</math> is mapped onto from at least one thing in <math>A</math>
  2 KB (263 words) - 21:56, 8 May 2018
• Dual vector space
  Given a [[Vector space|vector space]] {{M|(V,F)}} we define the '''dual''' or '''co * <math>V^*=\text{Hom}(V,F)</math> (recall this the set of all [[Homomorphism|homomorphisms]] (specifically [[Linear map|linear one
  3 KB (614 words) - 05:35, 8 December 2016
• Connected (topology)
  {{Refactor notice|grade=A|msg=Ancient page, needs an update, linking to theorems, so forth}} Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is ''connected'' if{{rITTMJML}}:
  5 KB (866 words) - 01:52, 1 October 2016
• Compactness
  ...I think there's a different definition for metric spaces, I have not seen a proof that the metric one {{M|\implies}} this one ...e may not speak of the compactness of subsets. Compactness is ''strictly'' a property of topological spaces.
  5 KB (828 words) - 15:59, 1 December 2015
• Subspace topology
  {{Refactor notice|grade=A|msg=Needed urgently, ready to plough on with it now though!}} ...y empty, possibly equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\mathcal{J}_S)}} where the [[topology]] {{M|\
  6 KB (1,146 words) - 23:04, 25 September 2016
• Quotient topology
  * Need to add [[Equivalent conditions to a map being a quotient map]] {{Refactor notice|grade=A|msg=This page is an embarrassment}}
  5 KB (795 words) - 13:34, 16 October 2016
• Motivation for topology
  ...from this definition. The [[Open ball|open balls]] are open sets, any open set is the union of open balls. Can we go further? <math>\forall\text{open sets}\in Y,\ f^{-1}(\text{that open set})</math> is open. This looks very different from the definition.
  1 KB (243 words) - 15:39, 13 February 2015
• Continuity definitions are equivalent
  ...th> being continuous (where the topologies are those [[Topology induced by a metric|induced by the metric]] are the same, that is ...all\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
• The relationship between logical implication and the subset relation
  ...h> (and we say "A is a subset of B") if and only if every element of <math>A</math> also belongs to <math>B</math> ...[x\in A\implies x\in B]</math><ref>Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jec
  776 B (136 words) - 17:36, 16 January 2017
• Basis for a topology
  {{Refactor notice|grade=A|msg=Updating with findings. * Need to add: [[A function is continuous if and only if the pre-image of every basis element
  5 KB (802 words) - 18:35, 17 December 2016