The separation axioms are additional conditions which may be required to a topological space in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases.
|given two distinct points, there is an open set containing exactly one of them;|
|(http://planetmath.org/T1Space)||given two distinct points, there is a neighborhood of each of them which does not contain the other point;|
|(http://planetmath.org/T2Space)||given two distinct points, there are two disjoint open sets each of which contains one of the points;|
|given two distinct points, there are two open sets, each of which contains one of the points, whose closures are disjoint;|
|(http://planetmath.org/T3Space)||given a closed set and a point , there are two disjoint open sets and such that and ;|
|given a closed set and a point , there is an Urysohn function for and ;|
|given two disjoint closed sets and , there are two disjoint open sets and such that and ;|
|given two separated sets and , there are two disjoint open sets and such that and .|
If a topological space satisfies a axiom, it is called a -space. The following table shows other common names for topological spaces with these or other additional separation properties.
|Completely Hausdorff space|
|Tychonoff or completely regular space||and|
|Perfectly space||and every closed set is a (see here (http://planetmath.org/G_deltaSet))|
|Perfectly normal space||and perfectly|
|Completely normal space||and|
The following implications hold strictly:
Remark. Some authors define spaces in the way we defined regular spaces, and spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.
Bibliography: Counterexamples in Topology, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)
|Date of creation||2013-03-22 13:28:47|
|Last modified on||2013-03-22 13:28:47|
|Last modified by||Koro (127)|