Hausdorff Space - Definitions

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 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.

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