Spaces of continuous functions, $\Sigma$-products and Box Topology

@article{Angoa2006,
abstract = {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 $C(X)$ denotes the space of real-valued continuous functions defined on $X$. $\square \mathbb \{R\}^\{\kappa \}$ is the Cartesian product $\mathbb \{R\}^\{\kappa \}$ with its box topology, and $C_\{\square \}(X)$ is $C(X)$ with the topology inherited from $\square \mathbb \{R\}^\{X\}$. By $\widehat\{C\}(X_1)$ we denote the set $\lbrace f\in C(X_1) : f$ can be continuously extended to all of $X\rbrace $. 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 intersection with the elements of an infinite subcollection of the given partition. We analyze $C_\square (X)$ when $X_0$ is $F_\sigma $ and prove: (1) for every topological space $X$, if $X_\{0\}$ is $F_\{\sigma \}$ in $X$, and $\emptyset \ne X_\{1\}\subset \operatorname\{cl\}_\{X\}X_\{0\}$, then $C_\{\square \}(X)\cong \square \mathbb \{R\}^\{X_\{0\}\}$; (2) for every space $X$ such that $X_\{0\}$ is $F_\{\sigma \}$, $\operatorname\{cl\}_\{X\}X_\{0\}\cap X_\{1\}\ne \emptyset $, and $X_1 \setminus \operatorname\{cl\}_X X_0$ is almost-$\omega $-resolvable, then $C_\{\square \}(X)$ is homeomorphic to a free topological sum of $\le |\widehat\{C\}(X_1)|$ copies of $\square \mathbb \{R\}^\{X_\{0\}\}$, and, in this case, $C_\{\square \}(X) \cong \square \mathbb \{R\}^\{X_\{0\}\}$ if and only if $|\widehat\{C\}(X_1)|\le 2^\{|X_\{0\}|\}$. We conclude that for a space $X$ such that $X_0$ is $F_\sigma $, $C_\square (X)$ is never normal if $|X_0| >\aleph _0$ [La], and, assuming CH, $C_\square (X)$ is paracompact if $|X_0| = \aleph _0$ [Ru2]. We also analyze $C_\square (X)$ when $|X_1| = 1$ and when $X$ is countably compact, and we scrutinize under what conditions $\square \mathbb \{R\}^\kappa $ is homeomorphic to some of its “$\Sigma $-products"; in particular, we prove that $\square \mathbb \{R\}^\omega $ is homeomorphic to each of its subspaces $\lbrace f \in \square \mathbb \{R\}^\omega : \lbrace n\in \omega : f(n) = 0\rbrace \in p\rbrace $ for every $p \in \omega ^*$, and it is homeomorphic to $\lbrace f \in \square \mathbb \{R\}^\omega : \,\, \forall \,\, \epsilon > 0 \,\, \lbrace n\in \omega : |f(n)| < \epsilon \rbrace \in \{\mathcal \{F\}\}_0\rbrace $ where $\mathcal \{F\}_0$ is the Fréchet filter on $\omega $.},
author = {Angoa, J., Tamariz-Mascarúa, Angel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {spaces of real-valued continuous functions; box topology; $\Sigma $-product; almost-$\omega $-resolvable space; spaces of real-valued continuous functions; -product; almost--resolvable space},
language = {eng},
number = {1},
pages = {69-94},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Spaces of continuous functions, $\Sigma $-products and Box Topology},
url = {http://eudml.org/doc/249848},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Angoa, J.
AU - Tamariz-Mascarúa, Angel
TI - Spaces of continuous functions, $\Sigma $-products and Box Topology
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 1
SP - 69
EP - 94
AB - 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 $C(X)$ denotes the space of real-valued continuous functions defined on $X$. $\square \mathbb {R}^{\kappa }$ is the Cartesian product $\mathbb {R}^{\kappa }$ with its box topology, and $C_{\square }(X)$ is $C(X)$ with the topology inherited from $\square \mathbb {R}^{X}$. By $\widehat{C}(X_1)$ we denote the set $\lbrace f\in C(X_1) : f$ can be continuously extended to all of $X\rbrace $. 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 intersection with the elements of an infinite subcollection of the given partition. We analyze $C_\square (X)$ when $X_0$ is $F_\sigma $ and prove: (1) for every topological space $X$, if $X_{0}$ is $F_{\sigma }$ in $X$, and $\emptyset \ne X_{1}\subset \operatorname{cl}_{X}X_{0}$, then $C_{\square }(X)\cong \square \mathbb {R}^{X_{0}}$; (2) for every space $X$ such that $X_{0}$ is $F_{\sigma }$, $\operatorname{cl}_{X}X_{0}\cap X_{1}\ne \emptyset $, and $X_1 \setminus \operatorname{cl}_X X_0$ is almost-$\omega $-resolvable, then $C_{\square }(X)$ is homeomorphic to a free topological sum of $\le |\widehat{C}(X_1)|$ copies of $\square \mathbb {R}^{X_{0}}$, and, in this case, $C_{\square }(X) \cong \square \mathbb {R}^{X_{0}}$ if and only if $|\widehat{C}(X_1)|\le 2^{|X_{0}|}$. We conclude that for a space $X$ such that $X_0$ is $F_\sigma $, $C_\square (X)$ is never normal if $|X_0| >\aleph _0$ [La], and, assuming CH, $C_\square (X)$ is paracompact if $|X_0| = \aleph _0$ [Ru2]. We also analyze $C_\square (X)$ when $|X_1| = 1$ and when $X$ is countably compact, and we scrutinize under what conditions $\square \mathbb {R}^\kappa $ is homeomorphic to some of its “$\Sigma $-products"; in particular, we prove that $\square \mathbb {R}^\omega $ is homeomorphic to each of its subspaces $\lbrace f \in \square \mathbb {R}^\omega : \lbrace n\in \omega : f(n) = 0\rbrace \in p\rbrace $ for every $p \in \omega ^*$, and it is homeomorphic to $\lbrace f \in \square \mathbb {R}^\omega : \,\, \forall \,\, \epsilon > 0 \,\, \lbrace n\in \omega : |f(n)| < \epsilon \rbrace \in {\mathcal {F}}_0\rbrace $ where $\mathcal {F}_0$ is the Fréchet filter on $\omega $.
LA - eng
KW - spaces of real-valued continuous functions; box topology; $\Sigma $-product; almost-$\omega $-resolvable space; spaces of real-valued continuous functions; -product; almost--resolvable space
UR - http://eudml.org/doc/249848
ER -