Difference between revisions of "Separation axioms"
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label("$A$",(A+C)/2,(2,-1)); label("$B$",(A+C)/2 + shiftfactor,(2,-1)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); | label("$A$",(A+C)/2,(2,-1)); label("$B$",(A+C)/2 + shiftfactor,(2,-1)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); | ||
</asy></center> | </asy></center> | ||
− | In a <math>T_4</math>, or a '''normal''', space, given any two disjoint closed sets <math>A,B</math> in a topological space <math>X</math>, there exists open sets <math>U,V</math> such that <math>A \subset U, B \ | + | In a <math>T_4</math>, or a '''normal''', space, given any two disjoint closed sets <math>A,B</math> in a topological space <math>X</math>, there exists open sets <math>U,V</math> such that <math>A \subset U, B \subset V</math> and <math>U,V</math> are [[disjoint]]. |
An example of a regular space that is not normal is the [[Sorgenfrey plane]]. | An example of a regular space that is not normal is the [[Sorgenfrey plane]]. |
Latest revision as of 12:28, 4 June 2018
The separation axioms are a series of definitions in topology that allow the classification of various topological spaces. The following axioms are typically defined: . Each axiom is a strictly stronger condition upon the topology than the previous axiom.
Contents
[hide]Accessible
In a , or an acessible, space, every one-point set is closed.
Hausdorff
In a , or an Hausdorff, space, given any two distinct points in a topological space , there exists open sets such that and are disjoint.
An example of a space that is but not is the finite complement topology on any infinite space.
Regular
In a , or a regular, space, given a point and a closed set in a topological space that are disjoint, there exists open sets such that and are disjoint.
An example of a Hausdorff space that is not regular is the space , the k-topology (or in more generality, a subspace of consisting of missing a countable number of elements).
Normal
In a , or a normal, space, given any two disjoint closed sets in a topological space , there exists open sets such that and are disjoint.
An example of a regular space that is not normal is the Sorgenfrey plane.
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