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.
@article{119357,
author = {Angel Tamariz-Mascar\'ua and H. Villegas-Rodr\'\i guez},
title = {Spaces of continuous functions, box products and almost-$\omega$-resolvable spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {687-705},
zbl = {1090.54011},
mrnumber = {2045790},
language = {en},
url = {http://dml.mathdoc.fr/item/119357}
}
Tamariz-Mascarúa, Angel; Villegas-Rodríguez, H. Spaces of continuous functions, box products and almost-$\omega$-resolvable spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 687-705. http://gdmltest.u-ga.fr/item/119357/
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