Fibrewise compactification

Of the many methods of compactification which are treated in the literature, two of the most important are the Wallman, of which the Stone–Čech may be regarded as a modification, and the Alexandroff (or one-point) compactification. We begin this second chapter by showing how to construct fibrewise versions of these two methods, which are very different in character, taking the Wallman method first.

As before we work over a topological base space B. Let X be fibrewise topological over B. We describe a subset M of X as b-closed, where b ∈ B, if Xw∩M is closed in Xw for some neighbourhood W of b. Thus it is sufficient, but not necessary, for M to be closed in X. For example Xw is b-closed for each neighbourhood W of b. Note that finite unions and finite intersections of b-closed sets are again b-closed.

By a b-closed filter on X, where b ∈ B, we mean a filter F on X consisting entirely of b-closed sets, which is tied to b in the sense that Xw ∈ F for each neighbourhood W of b. This is in line with the terminology used in §4 except that there the filters consist of unrestricted subsets of X. By a refinement of a b-closed filter F we mean a b-closed filter F′ such that each member of F is also a member of F′. By a b-closed ultrafilter we mean a b-closed filter which does not admit any strict refinement. When X is fibrewise R0-space as defined in (2.2), the filter consisting of all b-closed sets containing a given point x ∈ Xb is a b-closed ultrafilter, as can easily be seen.