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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 distinct points have disjoint neighbourhoods. 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. == Definitions == Points ''x'' and ''y'' in a topological space ''X'' 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 (). ''X'' is a Hausdorff space if any two distinct points of ''X'' can be separated by neighborhoods. 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. ''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 (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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hausdorff space」の詳細全文を読む スポンサード リンク
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