T1-locales

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Abstract
In the theory of locales (or ‘pointless topology'), several authors have tried to find a suitable form of a T1-separation axiom. Their proposals are unsatisfactory in the sense that they do not coincide for topological spaces with the T1-axiom. Our main result is that sublocales of sober T1-spaces are exactly locales in which primes are dual atoms. Hence, these locales should be called T1, despite the fact that they include any locale without points. Our T1-locales are also closed with respect to products; in fact they form the smallest epireflective subcategory of locales containing all sober T1-spaces. Besides T1-locales we will also consider T2-locales and epireflective sub-categories of locales in general.

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