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.

Perfect locales and localic real functions

“…For any perfect space X, the locale Ω(X) is perfect [7,Proposition 3.6]. For more information on localic perfectness we refer to [5,Section 4] and [7,Remarks 3.10].…”

“…Note that, in general, it is false that 0 < F iff 0 < ϕ F . A counterexample can be obtained from Proposition 3.3 of [7]: Let N be endowed with the cofinite topology and let F : N → R be given by…”

“…Actually, however, it is stronger (since S(L) is not Boolean in general). In [7], we called it the G δ -perfect property and observed that G δ -perfect and perfect properties do coincide under normality. It is also worth pointing that G δ -perfect sublocales of compact regular locales are spatial (see [15,VII.7.3]).…”

“…Combining normality and perfectness, we have the notion of a perfectly normal locale ( [5]). As mentioned before, this is the same as normal plus G δ -perfect (see [7] for the details). Then Theorem 5.2 plus Theorem 5.1 yield the pointfree version of Michael's insertion theorem formulated exactly as in [9] (see also [5]).…”

“…In the category of locales, these two formulations are no longer equivalent when phrased in terms of open sublocales and closed sublocales, in which case G δ -perfectness is generally stronger than F σ -perfectness. In [7], we took the weaker condition as the pointfree concept of perfectness and kept the terminology of a G δ -perfect locale for the locales that satisfy the stronger condition.…”