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Zbl 1212.54069
Angoa, J.; Ibarra, M.; Tamariz-Mascarúa, A.
On $\omega $-resolvable and almost-$\omega $-resolvable spaces. (English)
[J] Commentat. Math. Univ. Carol. 49, No. 3, 485-508 (2008). ISSN 0010-2628

Summary: We continue the study of almost-$\omega $-resolvable spaces started in [{\it A. Tamariz-Mascarúa} and {\it H. Villegas-Rodr\'{i}guez}, Commentat. Math. Univ. Carol. 43, No.~4, 687--705 (2002; Zbl 1090.54011)]. We prove in ZFC: (1) every crowded $T_0$ space with countable tightness and every $T_1$ space with $\pi $-weight $\leq \aleph _1$ is hereditarily almost-$\omega $-resolvable; (2) every crowded paracompact $T_2$ space which is the closed preimage of a crowded Fréchet $T_2$ space in such a way that the crowded part of each fiber is $\omega $-resolvable, has this property too; and (3) every Baire dense-hereditarily almost-$\omega $-resolvable space is $\omega $-resolvable. Moreover, by using the concept of almost-$\omega $-resolvability, we obtain two results the first one due to {\it O. Pavlov} and the other to {\it V. I. Malykhin}: (1) $V = L$ implies that every crowded Baire space is $\omega $-resolvable; and (2) $V = L$ implies that the product of two crowded spaces is resolvable. Finally, we prove that the product of two almost resolvable spaces is resolvable.

MSC 2000:: *54D10 Lower separation axioms
54E52 Baire category, Baire spaces
54A35 Consistency and independence results (general topology)
54C05 Continuous maps
54A10 Several topologies on one set

Keywords: Baire spaces; resolvable spaces; almost resolvable spaces; tightness; $\pi $-weight

Citations: Zbl 1090.54011

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Zentralblatt MATH Berlin [Germany]