Abstract:A dense-in-itself space $X$ is called {$C_\square $-discrete} if the space of real continuous functions on $X$ with its box topology, $C_\square (X)$, is a discrete space. A space $X$ is called {almost-$\omega $-resolvable} provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We analyze these classes of spaces by determining their relations with $\kappa $-resolvable and almost resolvable spaces. We prove that every almost-$\omega $-resolvable space is $C_\square $-discrete, and that these classes coincide in the realm of completely regular spaces. Also, we prove that almost resolvable spaces and almost-$\omega $-resolvable spaces are two different classes of spaces if there exists a measurable cardinal. Finally, we prove that it is consistent with $ZFC$ that every dense-in-itself space is almost-$\omega $-resolvable, and that the existence of a measurable cardinal is equiconsistent with the existence of a Tychonoff space without isolated points which is not almost-$\omega $-resolvable.
Keywords: box product, $\kappa $-resolvable space, almost resolvable space, almost-$\omega $-resolvable space, Baire irresolvable space, measurable cardinals
AMS Subject Classification: 54C35, 54F65, 54A35