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Union of Realcompact Spaces and Lindelöf Spaces

Published online by Cambridge University Press:  20 November 2018

Akio Kato
Akio Kato*
Affiliation:
National Defense Academy, Yokosuka, Japan
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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?

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 6 , 01 December 1979 , pp. 1247 - 1268
Copyright
Copyright © Canadian Mathematical Society 1979

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