# Preregular

### Some articles on preregular:

Hausdorff Space - Preregularity Versus Regularity
... All regular spaces are preregular, as are all Hausdorff spaces ... Most of the time, these results hold for all preregular spaces they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came ... also (say) locally compact will be regular, because any Hausdorff space is preregular ...
Hausdorff Space - 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 ... up to topological indistinguishability (for preregular spaces) ... and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T0 condition ...
Hausdorff Space - Examples and Counterexamples
... 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 ... 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 abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring ...
Hausdorff Space - Definitions
... 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 ...
Regular Space - Relationships To Other Separation Axioms
... A regular space is necessarily also preregular, i.e ... Since a Hausdorff space is the same as a preregular T0 space, a regular space that is also T0 must be Hausdorff (and thus T3) ... Most of the time, these results hold for all preregular spaces they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later ...