Directed inverse limits of spatial locales

Abstract

Abstract. In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.The question "Which limits are preserved by the embedding of sober topological spaces into the category of locales?", or equivalently "When is a limit of spatial locales spatial?" seems to be rather difficult. Results concerning preservation of products (and some applications) can be found in [2,7,8]. Here we consider preservation of directed inverse limits; in particular Isbell's question whether a directed inverse limit of (quasi-)compact spatial locales is again spatial [1, p. 24]. It is easy to see that if all locales are Hausdorff (i.e. if they have closed diagonals), then this question has a positive answer because (i) compact Hausdorff locales are spatial and closed under products, (ii) limits can be constructed as equalizers of a pair of maps between products of the original objects, and (iii) equalizers of maps between Hausdorff locales are closed inclusions and compactness is inherited by closed sublocales. So only the case of non-Hausdorff locales is problematic.This note consists of two parts. In the first part we give a construction which shows that directed inverse limits of compact locales need not be spatial even if all locales are furthermore locally compact and all maps open. However, in the second part we show that if all locales are locally compact and all maps proper (a weaker property suffices), then the inverse limit is again locally compact and hence spatial; here compactness is not needed.Constructions of the limit of a directed inverse system [6] or [10]. We don't need to go into any detail here since we will only be using the fact that if {B j } is a collection of bases for the locales X j , then the union of the inverse images under the maps f * ∞j is a base for the limit. This union is a subbase for any limit. But directedness implies that it is in fact a base.