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In Topology and related branches of Mathematics , a Hausdorff space, '''separated space''' or '''T2 space''' is a Topological Space in which points can be ''separated by neighbourhoods''. Of the many Separation Axiom s that can be imposed on a topological space, the Hausdorff condition is the most frequently used and discussed. It implies the uniqueness of Limit s of Sequence s, Net s, and Filter s. Hausdorff spaces are named for Felix Hausdorff , one of the founders of topology. Hausdorff's original definition of a topological space included the Hausdorff condition as an axiom. DEFINITIONS Suppose that ''X'' is a Topological Space . Let ''x'' and ''y'' be Points in ''X''. We say that ''x'' and ''y'' can be '' Separated By Neighbourhoods '' if There Exists a Neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''y'' such that ''U'' and ''V'' are Disjoint (''U'' ∩ ''V'' = ∅). ''X'' is a Hausdorff space if any two Distinct points of ''X'' can be separated by neighborhoods. This is why Hausdorff spaces are also called ''T2 spaces'' or ''separated spaces''. ''X'' is a preregular space if any two Topologically Distinguishable points can be separated by neighbourhoods. Preregular spaces are also called ''R1 spaces''. The relationship between these two conditions is as follows. A topological space is Hausdorff If And Only If it is both preregular and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov Quotient is Hausdorff. EQUIVALENCES For a topological space ''X'', the following are equivalent: |
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