Separation axioms: Difference between revisions
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In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s of the topology distinguish between distinct points. | In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s of the topology distinguish between distinct points. | ||
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A ''neighbourhood of a set'' ''A'' in ''X'' is a set ''N'' such that ''A'' is contained in the interior of ''N''; that is, there is an open set ''U'' such that <math>A \subseteq U \subseteq N</math>. | A ''neighbourhood of a set'' ''A'' in ''X'' is a set ''N'' such that ''A'' is contained in the interior of ''N''; that is, there is an open set ''U'' such that <math>A \subseteq U \subseteq N</math>. | ||
Subsets ''U'' and ''V'' are ''separated'' in ''X'' if ''U'' is disjoint from the [[Closure (topology)|closure]] of ''V'' and ''V'' is disjoint from the closure of ''U''. | |||
== | A '''Urysohn function''' for subsets ''A'' and ''B'' of ''X'' is a [[continuous function]] ''f'' from ''X'' to the real unit interval such that ''f'' is 0 on ''A'' and 1 on ''B''. | ||
==Axioms== | |||
A topological space ''X'' is | A topological space ''X'' is | ||
* '''T0''' if for any two distinct points there is an open set which contains just one | * '''T0''' if for any two distinct points there is an open set which contains just one | ||
* '''T1''' if for any two points ''x'', ''y'' there are open sets ''U'' and ''V'' such that ''U'' contains ''x'' but not ''y'', and ''V'' contains ''y'' but not ''x'' | * '''T1''' if for any two points ''x'', ''y'' there are open sets ''U'' and ''V'' such that ''U'' contains ''x'' but not ''y'', and ''V'' contains ''y'' but not ''x'' | ||
* '''T2''' if any two distinct points have disjoint neighbourhoods | * '''T2''' if any two distinct points have disjoint neighbourhoods | ||
* '''T2½''' if distinct points have disjoint closed neighbourhoods | |||
* '''T3''' if a closed set ''A'' and a point ''x'' not in ''A'' have disjoint neighbourhoods | * '''T3''' if a closed set ''A'' and a point ''x'' not in ''A'' have disjoint neighbourhoods | ||
* '''T3½''' if for any closed set ''A'' and point ''x'' not in ''A'' there is a Urysohn function for ''A'' and {''x''} | |||
* '''T4''' if disjoint closed sets have disjoint neighbourhoods | * '''T4''' if disjoint closed sets have disjoint neighbourhoods | ||
* '''T5''' if separated sets have disjoint neighbourhoods | * '''T5''' if separated sets have disjoint neighbourhoods | ||
* '''Hausdorff''' is a synonym for T2 | * '''Hausdorff''' is a synonym for T2 | ||
* ''' | * '''completely Hausdorff''' is a synonym for T2½ | ||
* '''regular''' if T0 and T4 | |||
* '''regular''' if T0 and T3 | |||
* '''completely regular''' if T0 and T3½ | |||
* '''Tychonoff''' is completely regular and T1 | |||
* '''normal''' if T0 and T4 | |||
* '''completely normal''' if T1 and T5 | |||
* '''perfectly normal''' if normal and every closed set is a [[G-delta set|G<sub>δ</sub>]] | |||
==Properties== | |||
* A space is T1 if and only if each point ([[singleton]]) forms a closed set. | |||
* ''Urysohn's Lemma'': if ''A'' and ''B'' are disjoint closed subsets of a T4 space ''X'', there is a Urysohn function for ''A'' and ''B'''. | |||
==References== | ==References== | ||
* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 }} | * {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:01, 17 October 2024
In topology, separation axioms describe classes of topological space according to how well the open sets of the topology distinguish between distinct points.
Terminology
A neighbourhood of a point x in a topological space X is a set N such that x is in the interior of N; that is, there is an open set U such that . A neighbourhood of a set A in X is a set N such that A is contained in the interior of N; that is, there is an open set U such that .
Subsets U and V are separated in X if U is disjoint from the closure of V and V is disjoint from the closure of U.
A Urysohn function for subsets A and B of X is a continuous function f from X to the real unit interval such that f is 0 on A and 1 on B.
Axioms
A topological space X is
- T0 if for any two distinct points there is an open set which contains just one
- T1 if for any two points x, y there are open sets U and V such that U contains x but not y, and V contains y but not x
- T2 if any two distinct points have disjoint neighbourhoods
- T2½ if distinct points have disjoint closed neighbourhoods
- T3 if a closed set A and a point x not in A have disjoint neighbourhoods
- T3½ if for any closed set A and point x not in A there is a Urysohn function for A and {x}
- T4 if disjoint closed sets have disjoint neighbourhoods
- T5 if separated sets have disjoint neighbourhoods
- Hausdorff is a synonym for T2
- completely Hausdorff is a synonym for T2½
- regular if T0 and T3
- completely regular if T0 and T3½
- Tychonoff is completely regular and T1
- normal if T0 and T4
- completely normal if T1 and T5
- perfectly normal if normal and every closed set is a Gδ
Properties
- A space is T1 if and only if each point (singleton) forms a closed set.
- Urysohn's Lemma: if A and B are disjoint closed subsets of a T4 space X, there is a Urysohn function for A and B'.
References
- Steen, Lynn Arthur & J. Arthur Jr. Seebach (1978), Counterexamples in Topology, Berlin, New York: Springer-Verlag, ISBN 0-387-90312-7