# Article

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Keywords:
Baire spaces; resolvable spaces; almost resolvable spaces; almost-$\omega$-resolvable spaces; tightness; $\pi$-weight
Summary:
We continue the study of almost-$\omega$-resolvable spaces beginning in A. Tamariz-Mascar'ua, H. Villegas-Rodr'{\i}guez, {\it Spaces of continuous functions, box products and almost-$\omega$-resoluble spaces\/}, Comment. Math. Univ. Carolin. {\bf 43} (2002), no. 4, 687--705. 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 due the first one to O. Pavlov and the other to 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.
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