Hausdorff space

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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. Intuitively, the condition is illustrated by the pun that a space is Hausdorff if any two points can be "housed off" from each other by open sets.[1]

Hausdorff spaces are named for 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.

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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 (UV = ∅). 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) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

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