Geodesic
Union of Realcompact Spaces and Lindelöf Spaces
Canadian journal of mathematics, Tome 31 (1979) no. 6, pp. 1247-1268

Voir la notice de l'article provenant de la source Cambridge University Press

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? 
Kato, Akio. Union of Realcompact Spaces and Lindelöf Spaces. Canadian journal of mathematics, Tome 31 (1979) no. 6, pp. 1247-1268. doi: 10.4153/CJM-1979-104-8
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