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For a Tychonoff space $X$, we will denote by $X_0$ the set of its isolated points and $X_1$ will be equal to $X\setminus X_0$. The symbol $C\left(X\right)$ denotes the space of real-valued continuous functions defined on $X$. $\square {ℝ}^{\kappa }$ is the Cartesian product ${ℝ}^{\kappa }$ with its box topology, and $C_{\square }\left(X\right)$ is $C\left(X\right)$ with the topology inherited from $\square {ℝ}^X$. By $\widehat{C}\left(X_1\right)$ we denote the set $\{f\in C\left(X_1\right):f$ can be continuously extended to all of $X\}$. A space $X$ is almost-$\omega$-resolvable if it can be partitioned by a countable family of subsets in such a way that every non-empty open subset of $X$ has a non-empty...

We continue the study of almost-$\omega$-resolvable spaces beginning in A. Tamariz-Mascar’ua, H. Villegas-Rodr’ıguez, , Comment. Math. Univ. Carolin. (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 $\le {\aleph }_1$ 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)...