### Some articles on *preregular*:

Hausdorff Space -

... All regular spaces are

**Preregular**ity 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 "

... 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

... 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

... 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

... 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 ...Main Site Subjects

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