Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exists a neighbourhood of each which is disjoint from the neighbourhood of the 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. In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exists a neighbourhood of each which is disjoint from the neighbourhood of the 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 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 (U ∩ V = ∅).X is a Hausdorff space if all distinct points in X are pairwise neighbourhood-separable. 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 disjoint 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. For a topological space X, the following are equivalent: Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.