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Keywords:
box product; $\kappa$-resolvable space; almost resolvable space; almost-$\omega$-resolvable space; Baire irresolvable space; measurable cardinals
box product; $\kappa$-resolvable space; almost resolvable space; almost-$\omega$-resolvable space; Baire irresolvable space; measurable cardinals
Summary:
A dense-in-itself space $X$ is called {\it $C_\square$-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 {\it 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.
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