Perfectness in locales

Abstract

This paper makes a comparison between two notions of perfectness for locales which come as direct reformulations of the two equivalent topological definitions of perfectness. These reformulations are no longer equivalent. It will be documented that a locale may appropriately be called perfect if each of its open sublocales is a join of countably many closed sublocales. Certain circumstances are exhibited in which both reformulations coincide. This paper also studies perfectness in mildly normal locales. It is shown that perfect and mildly normal locales coincide with the Oz locales extensively studied in the last decade.

Mathematics Subject Classication (2010): 06D22, 54D15.

Key words: Locale, sublocale, Fα-sublocale, Gα-sublocale, normality, mild normality,
perfectness, perfect normality, pm-normality, Oz locale.