Union of Realcompact Spaces and Lindelöf Spaces

Abstract

All spaces in this paper are completely regular Hausdorff and all maps are continuous onto, unless otherwise stated. The purpose of this paper is to investigate the realcompactness of a space X which contains a Lindelöf space L such that every zero-set Z (in X) disjoint from L is realcompact. We show in § 2 that such a space X is very close to being realcompact (Theorems I, II and III). But in general such a space fails to be realcompact. Indeed, in §§ 3 and 4 the following questions of Mrówka [18, 19] are answered, both in the negative:(Q. 1) If X = L ∪ G where L is Lindelöf closed and G is E-compact, then is X E-compact?(Q. 2) Suppose f:X → Y is a perfect map such that the set M(f) = {y ∊ Y| |f−l(y)| > 1} of multiple points of f is Lindelöf (especially, countable) closed. If X is E-compact, is Y also E-compact?

Topological measure theory for completely regular spaces and their projective covers

“…These examples emphasize the almost realcompact spaces introduced by Frolik [10] and recently studied by Kato [18]. The notion of a weak cb space, introduced by Mack and Johnson [25] in their study of the lattice completion of C(X), is characterized in a new way, leading to the principal positive result of the paper: If X is weak cb, then X has the SLP.…”

“…A space is almost realcompact [10] if every ultrafilter of regular closed sets such that countable subfamilies have nonempty intersection is fixed. It is known that realcompact implies almost realcompact, but not conversely [18]. Almost realcompactness is preserved by closed subsets and by products, from [14], every X admits an "almostrealcompactification" aX, with XczaXavXczβX.…”

Extension properties induced by complete quasi-uniformities

“…The remaining implication of the proposition is an immediate consequence of the well-known result of E. e ω x X ω} is a subbase. It is well known that X is almost realcompact [27]. To see that X is not LF-complete, let Φ be an ultrafilter on X containing {(α, ω λ ) X [n, ω): α€ω l9 ft€ω}.…”

On unions of 𝜈-embedded sets

“…Next, we shall show that in Theorem 2.3 "C-embedded" cannot be weakened to "C*-embedded". The following example is a modification of the Dieudonné plank in [3]. Example 2.6.…”