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Hausdorff spaces are named for Felix Hausdorff , one of the founders of topology. In fact, 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. EXAMPLES AND COUNTEREXAMPLES Almost all spaces encountered in Analysis are Hausdorff; most importantly, the Real Number s are a Hausdorff space. More generally, all Metric Space s are Hausdorff. In fact, many spaces of use in analysis, such as Topological Group s and Topological Manifold s, 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 . Pseudometric Space s typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff Gauge Space s. 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. In contrast, non-preregular spaces are encountered much more frequently in Heyting Algebra is the algebra of Open Set s of some topological space, but this space need not be preregular, much less Hausdorff. PROPERTIES One of the nicest properties of Hausdorff spaces is that Limit s of Sequence s, Net s, and Filter s are unique whenever they exist. In fact, a topological space is Hausdorff if and only if every net (or filter) has at most one limit. Similarly, a space is preregular if all of the limits of a given net (or filter) are topologically indistinguishable. |
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