Abstract:We continue the study of almost-$\omega $-resolvable spaces beginning in A. Tamariz-Mascar\'ua, H. Villegas-Rodr\'{\i }guez, {Spaces of continuous functions, box products and almost-$\omega $-resoluble spaces}, Comment. Math. Univ. Carolin. {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 Fr\'echet $T_2$ 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.
Keywords: Baire spaces, resolvable spaces, almost resolvable spaces, almost-$\omega $-resolvable spaces, tightness, $\pi $-weight
AMS Subject Classification: Primary 54D10, 54E52, 54A35; Secondary 54C05, 54A10