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.
In a , or an acessible, space, every one-point set is closed.
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.
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).
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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