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  • Topology
    ...but rather a [[topological space]] which is a topology with its underlying set. See that page for more details}} ...}(X))}} if you prefer, here {{M|\mathcal{P}(X)}} denotes the [[power-set]] of {{M|X}}. This means that if {{M|U\in\mathcal{J} }} then {{M|U\subseteq X}}<
    3 KB (543 words) - 09:28, 30 December 2016
  • Open set
    ...} is an open set. {{M|\mathcal{J} }} is by definition the set of open sets of {{M|X}} ...l be shown that they are equivalent. Here {{M|U}} is some arbitrary subset of {{M|X}}.
    4 KB (677 words) - 02:26, 29 November 2015
  • Connected (topology)
    </ref> (expand the yellow box below for a reminder of this definition) '''Recall''' the definition of a topological space being ''{{link|disconnected|topology}}''
    5 KB (866 words) - 01:52, 1 October 2016
  • Compactness
    There are 2 distinct definitions of compactness, however they are equivalent: ...peak of the compactness of subsets. Compactness is ''strictly'' a property of topological spaces.
    5 KB (828 words) - 15:59, 1 December 2015
  • Subspace topology
    ...\iff S\subseteq X}}, so it's another way of saying let {{M|S}} be a subset of {{M|X}}, possibly empty, possibly equal to {{M|X}} itself</ref> be given. W ...ets of {{M|(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
  • Algebra of sets
    {{:Algebra of sets/Infobox}} : '''Note: ''' Every ''algebra of sets'' is a ''[[ring of sets]]'' (see below)
    3 KB (507 words) - 18:43, 1 April 2016
  • Sigma-ring
    ...and just like [[Algebra of sets|algebras of sets]] are related to rings of sets, so is <math>\sigma</math>-ring to [[Sigma-algebra|<math>\sigma</math>-alge A non-empty class of sets {{M|S}} is a {{sigma|ring}} if<ref>Measure Theory, p24 - Halmos - Graduate
    728 B (125 words) - 15:34, 13 March 2015
  • Sigma-algebra
    ...or {{sigma|algebra}} is very similar to a [[Sigma-ring|{{sigma|ring}}]] of sets. :: A [[ring of sets]] is to an [[algebra of sets]] as a [[sigma-ring|{{sigma|ring}}]] is to a ''{{sigma|algebra}}''
    8 KB (1,306 words) - 01:49, 19 March 2016
  • Measure Theory
    * [[Ring of sets]] * [[Algebra of sets]]
    4 KB (733 words) - 01:41, 28 March 2015
  • Pre-measure
    ...ales - Rene L. Schilling</ref> is a [[Measure|measure]] on an [[Algebra of sets|algebra]] rather than a [[Sigma-algebra|{{Sigma|algebra}}]], the properties * Here {{M|\mathcal{A} }} is an algebra of sets (a system of subsets of {{M|X}}) and {{M|\mu_0:\mathcal{A}\rightarrow[0,+\infty]}} such that:
    5 KB (782 words) - 01:49, 26 July 2015
  • Open and closed maps
    * The image of an open set is open (that is <math>\forall U\in\mathcal{J}[f(U)\in\mathcal{K}]</math>) ===Closed map===
    4 KB (692 words) - 08:00, 8 April 2015
  • Relatively open
    ...Topological space|topological space]] {{M|(X,\mathcal{J})}}, the open sets of {{M|(Y,\mathcal{J}_\text{subspace})}} are said to be '''relatively open'''< Alternatively we may say given a {{M|A\subseteq X}} the family of sets:
    592 B (97 words) - 18:42, 19 April 2015
  • Continuity and non-surjective functions
    ...\cdot\vert)]}} where {{M|\mathcal{O} }} denotes the [[open set|open sets]] of a space ...ce, and now take any open set that intersects with that path. A good chunk of that isn't used, what if it overlaps with 2 or more line segments?
    3 KB (556 words) - 17:42, 6 September 2015
  • Types of set algebras
    ...being refactored after being unchanged for more than 14 months. the order of elements and subheadings are likely to change and elements will be moved to ...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
  • Semi-ring of half-closed-half-open intervals
    ...bers</ref><ref group="Note">The symbol {{M|\subset}} could be used instead of {{M|\subseteq}} but it doesn't matter, as: ...mely {{M|n\in\mathbb{N} }}), we define {{M|1=\[a,b\)}}, a ''half-open-half-closed rectangle in {{M|\mathbb{R}^n}}''{{rMIAMRLS}} as follows:
    4 KB (680 words) - 00:23, 20 August 2016
  • Class of sets closed under complements properties
    If {{M|\mathcal{A} }} is a system of subsets of {{M|\Omega}} such that<ref name="PTACC">Probability Theory - A comprehensiv ...Complement|complement]] of {{M|A}} - That is to say "{{M|\mathcal{A} }} is closed under complements"
    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
  • Notes:Just what is in a generated sigma-algebra
    ...plies and subset relation|implies-subset relation]]) every [[Open set|open set]] is in {{M|\sigma(\mathcal{J})}}. We also know that {{M|\sigma(\mathcal{J} ...ations a [[Sigma-algebra|{{Sigma|algebra}}]] allows starting from the open sets, {{M|\mathcal{J} }}
    1 KB (256 words) - 13:29, 17 June 2015
  • Site projects:Patrolling measure theory/Task list
    * [[Algebra of sets]] - '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 18:45, 1 April * [[Class of sets closed under set-subtraction properties]] - '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|
    5 KB (645 words) - 11:40, 21 August 2016
  • Borel sigma-algebra generated by
    ...ure theory this notation is often used to denote the set of half-open-half-closed rectangles in {{M|\mathbb{R}^n}} - a totally separate thing</ref> is any [[ ! {{M|1=\mathcal{B}(X)=\sigma(\mathcal{C})}} - the closed sets
    2 KB (244 words) - 08:30, 6 August 2015

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