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Concept
Hausdorff space
Summary
In topology and related branches of mathematics, a Hausdorff space (ˈhaʊsdɔːrf , ˈhaʊzdɔːrf ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.
Points and in a topological space can be separated by neighbourhoods if there exists a neighbourhood of and a neighbourhood of such that and are disjoint . is a Hausdorff space if any two distinct points in are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used.
A related, but weaker, notion is that of a preregular space. is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. A preregular space is also called an R1 space.
The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.
For a topological space , the following are equivalent:
is a Hausdorff space.
Limits of nets in are unique.
Limits of filters on are unique.
Any singleton set is equal to the intersection of all closed neighbourhoods of . (A closed neighbourhood of is a closed set that contains an open set containing .)
The diagonal is closed as a subset of the product space .
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