Spaces of continuous functions, box products and almost- ω -resolvable spaces

Angel Tamariz-Mascarúa; H. Villegas-Rodríguez

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 4, page 687-705
  • ISSN: 0010-2628

Abstract

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A dense-in-itself space X is called C □ -discrete if the space of real continuous functions on X with its box topology, C □ ( X ) , is a discrete space. A space X is called almost- ω -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 κ -resolvable and almost resolvable spaces. We prove that every almost- ω -resolvable space is C □ -discrete, and that these classes coincide in the realm of completely regular spaces. Also, we prove that almost resolvable spaces and almost- ω -resolvable spaces are two different classes of spaces if there exists a measurable cardinal. Finally, we prove that it is consistent with Z F C that every dense-in-itself space is almost- ω -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- ω -resolvable.

How to cite

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Tamariz-Mascarúa, Angel, and Villegas-Rodríguez, H.. "Spaces of continuous functions, box products and almost-$\omega $-resolvable spaces." Commentationes Mathematicae Universitatis Carolinae 43.4 (2002): 687-705. <http://eudml.org/doc/248977>.

@article{Tamariz2002,
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.},
author = {Tamariz-Mascarúa, Angel, Villegas-Rodríguez, H.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {box product; $\kappa $-resolvable space; almost resolvable space; almost-$\omega $-resolvable space; Baire irresolvable space; measurable cardinals; box product topology; almost--resolvable space},
language = {eng},
number = {4},
pages = {687-705},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Spaces of continuous functions, box products and almost-$\omega $-resolvable spaces},
url = {http://eudml.org/doc/248977},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Tamariz-Mascarúa, Angel
AU - Villegas-Rodríguez, H.
TI - Spaces of continuous functions, box products and almost-$\omega $-resolvable spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 687
EP - 705
AB - 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.
LA - eng
KW - box product; $\kappa $-resolvable space; almost resolvable space; almost-$\omega $-resolvable space; Baire irresolvable space; measurable cardinals; box product topology; almost--resolvable space
UR - http://eudml.org/doc/248977
ER -

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