preregular

Adjective

Meaning

preregular (not comparable)

(topology, of a space) In which any two topologically distinguishable points can be separated by neighbourhoods

Examples

Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Source: Internet

Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Source: Internet

As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T 0 condition. Source: Internet

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). Source: Internet

A related, but weaker, notion is that of a preregular space. Source: Internet

As described above, any completely regular space is regular, and any T 0 space that is not Hausdorff (and hence not preregular) cannot be regular. Source: Internet

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