Hausdorff Space - Variants

Variants

The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces).

As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only if every Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).

Read more about this topic:  Hausdorff Space

Other articles related to "variants":

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... and low fuel capacity of the MiG-21F, PF, PFM, S/SM and M/MF variants—though each had a somewhat greater fuel capacity than its predecessor—led to the development of the MT and SMT variants ... Its G-limits were increased from +7Gs in initial variants to +8.5Gs in the latest variants ...
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Famous quotes containing the word variants:

    Nationalist pride, like other variants of pride, can be a substitute for self-respect.
    Eric Hoffer (1902–1983)