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• The intersection of sets is a subset of each set
  That <math>A\cap B\subset A</math> ...h>A\cap B\subset B</math> (as <math>A\cap B=B\cap A</math> and <math>B\cap A\subset B</math> by the statement above)<ref>Alec's (my) own work</ref>
  820 B (159 words) - 21:35, 15 August 2015
• Commutativity of intersection
  <math>A\cap B=B\cap A</math> This is somewhere between a theorem and a definition because at some point you have to accept "and" is commutative or
  610 B (122 words) - 19:32, 28 October 2016
• Closed set
  ...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</re ...y [[metric space]] is also a [[topological space]] it is still true that a set is closed if its complement is open.
  1 KB (238 words) - 15:36, 24 November 2015
• Vector space
  A vector space {{M|V}} over a [[Field|field]] {{M|F}} is a non empty set {{M|V}} and the binary operations: 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
  ...n Apr 2015 when [[Template:Theorem]] was moved to [[Template:Theorem Of]]. A very old page indeed!}} ...et {{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }} be a [[metric]] on that set and let {{M|(X,d)}} be the resulting [[metric space]]. Then we claim:
  4 KB (814 words) - 22:16, 16 January 2017
• Closure, interior and boundary
  ...} }} ("bar" in LaTeX) or {{M|\overline{A} }} ("overline" in LaTeX) is the set:<ref>Introduction to Topological Manifolds - John Lee</ref> ...h>\overline{A}=\bigcap\{B\subset X|A\subset B\text{ and }B\text{ is closed in }X\}</math>
  1 KB (210 words) - 00:20, 9 March 2015
• Measure space
  ...ame="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]]: * {{M|(X,\mathcal{A},\mu:\mathcal{A}\rightarrow[0,+\infty])}} - but because [[Mathematicians are lazy]] we simp
  1 KB (188 words) - 15:24, 21 July 2015
• Triangle inequality
  ...[Metric space|metric space]]) of which <math>|x-z|\le|x-y|+|y-z|</math> is a special case. ...way of writing it is <math>|a+b|\le |a|+|b|</math>, notice if we set {{M|1=a=x-y}} and {{M|1=b=y-z}} then we get <math>|x-y+y-z|\le|x-y|+|y-z|</math> wh
  3 KB (546 words) - 13:05, 19 February 2016
• Product topology
  {{Refactor notice|grade=A|As a part of the topology patrol}} {{Requires references|grade=A|msg=Check Munkres and Topological Manifolds}}
  5 KB (871 words) - 20:32, 23 September 2016
• Tuple
  ...ever as it may not be in a [[Vector space|vector space]] we do not call it a vector. ...3,\{a\})</math> is the tuple of the numbers 1, 2 and 3 and the set <math>\{a\}</math>
  610 B (97 words) - 16:30, 23 August 2015
• Function
  A function {{M|f}} is a special kind of [[Relation|relation]] ...]<ref name="API">Analysis - Part 1: Elements - Krzysztof Maurin</ref>, for a relation:
  4 KB (659 words) - 13:01, 19 February 2016
• Set theory axioms
  ...sed to provide some discussion for the axioms (for example "there exists a set with no elements" doesn't really deserve its own page) ...lt. Only "major" results are shown, they are covered in the motivation for set theory page, and "D" denotes "definition" - which is something that is natu
  3 KB (619 words) - 10:25, 11 March 2015
• Motivation for set theory axioms
  ...s. For example you are happy that {{M|\{1,2,3\}=\{2,1,3\} }} and that this set (I use singular because they are identical) contains the elements {{M|1}}, ...- for example we use a letter (<math>\emptyset</math>) to denote the empty set, what if there are 2 empty sets? We would like to justify this.
  3 KB (584 words) - 23:03, 28 February 2015
• Relation
  ...should be assumed if just relation is specified</ref>) between two sets is a subset of the [[Cartesian product]] of two sets{{rAPIKM}}<ref name="TAPL">T 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
• Ordered pair
  ...dered pair <math>(a,b)=\{\{a\},\{a,b\}\}</math>, this way <math>(a,b)\ne(b,a)</math>. Ordered pairs are vital in the study of [[Relation|relations]] which leads to [[Function|functions]]
  2 KB (327 words) - 07:22, 27 April 2015
• The domain, range and field of a relation exist
  It is important to know that the domain and range of a [[Relation|relation]] exist. ...math>\{x|\exists y:(x,y)\in R\}</math> exists, to do this we require the [[Set theory axioms|axioms]] of schema and union.
  339 B (63 words) - 07:22, 27 April 2015
• Ordering
  ...g|Proof that <math>\le</math> is a partial ordering <math>\iff <</math> is a strict ordering]] ...special kind of [[Relation|relation]], we can define an order uniquely as a partial or strict ordering. That is the two are equivalent.
  5 KB (1,006 words) - 13:21, 1 January 2016
• Equivalence relation
  A [[relation]], {{M|\sim}}, in {{M|X}}<ref group="Note">This terminology means {{M|\sim \subseteq X\times | {{M|\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
• Linear map
  ...a u+\beta v)=\alpha T(u)+\beta T(v)\ \forall u,v\in V\ \forall\alpha,\beta\in F}} | A linear transform into the same space as the domain, that is {{M|T:(V,F)\rig
  3 KB (512 words) - 16:30, 23 August 2015
• Index of notation
  ...b],\mathbb{R})</math> denotes the continuous function on the interval {{M|[a,b]}} that map to {{M|\mathbb{R} }} - this is unlikely to be given any other ! [[Index of set-like notations]]
  9 KB (1,490 words) - 06:13, 1 January 2017