>>12969611Beyond topological spaces
Hausdorff locales
The most obvious definition for a locale XX to be Hausdorff is that its diagonal XX×XX\to X\times X is a closed (and hence proper) inclusion. However, if XX is a sober space regarded as a locale, this might not coincide with the condition for XX to be Hausdorff as a space, since the Cartesian product X×XX\times X in the category Loc of locales might not coincide with the product in the category Top of topological spaces (the Tychonoff product). But it does coincide if XX is a locally compact locale, so in that case the two notions of Hausdorff are the same.
Separated toposes and schemes
This notion of a Hausdorff locale is a special case of that of Hausdorff topos in topos theory. This also is formally similar to notions such as a separated scheme etc. The corresponding relative notion (over an arbitrary base topos) is that of separated geometric morphism. For schemes see separated morphism of schemes.
7. In constructive mathematics
In constructive mathematics, the Hausdorff notion multifurcates further, due to the variety of possible meanings of closed subspace. If we ask the diagonal to be weakly closed, then in the spatial case, this means that it contains all its limit points, giving Definition 2.2 above. But if we ask the diagonal to be strongly closed, i.e. the complement of an open set, then in the spatial case this means that there is a tight inequality ?\ne (the exterior of ==) relative to which Definition 2.1 holds. (We use ?\ne twice in that definition: in the hypothesis that x?yx \ne y and in the conclusion that x?yx' \ne y'.)
It is natural to call these conditions weakly Hausdorff and strongly Hausdorff, but one should be aware of terminological clashes: in classical mathematics there is a different notion of a weak Hausdorff space, whereas (strong) Hausdorffness for locales has by some authors been called “strongly Hausdorff” only to contrast it with Hausdorffness for spaces.
