the main separation axioms
number | name | statement | reformulation |
---|---|---|---|
Kolmogorov | given two distinct points, at least one of them has an open neighbourhood not containing the other point | every irreducible closed subset is the closure of at most one point | |
given two distinct points, both have an open neighbourhood not containing the other point | all points are closed | ||
Hausdorff | given two distinct points, they have disjoint open neighbourhoods | the diagonal is a closed map | |
and… | all points are closed and… | ||
regular Hausdorff | …given a point and a closed subset not containing it, they have disjoint open neighbourhoods | …every neighbourhood of a point contains the closure of an open neighbourhood | |
normal Hausdorff | …given two disjoint closed subsets, they have disjoint open neighbourhoods | …every neighbourhood of a closed set also contains the closure of an open neighbourhood … every pair of disjoint closed subsets is separated by an Urysohn function |
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