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

• Volume: 47, Issue: 1, page 69-94
• ISSN: 0010-2628

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## Abstract

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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 $C\left(X\right)$ denotes the space of real-valued continuous functions defined on $X$. $\square {ℝ}^{\kappa }$ is the Cartesian product ${ℝ}^{\kappa }$ with its box topology, and ${C}_{\square }\left(X\right)$ is $C\left(X\right)$ with the topology inherited from $\square {ℝ}^{X}$. By $\stackrel{^}{C}\left({X}_{1}\right)$ we denote the set $\left\{f\in C\left({X}_{1}\right):f$ can be continuously extended to all of $X\right\}$. 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 ${C}_{\square }\left(X\right)$ when ${X}_{0}$ is ${F}_{\sigma }$ and prove: (1) for every topological space $X$, if ${X}_{0}$ is ${F}_{\sigma }$ in $X$, and $\varnothing \ne {X}_{1}\subset {cl}_{X}{X}_{0}$, then ${C}_{\square }\left(X\right)\cong \square {ℝ}^{{X}_{0}}$; (2) for every space $X$ such that ${X}_{0}$ is ${F}_{\sigma }$, ${cl}_{X}{X}_{0}\cap {X}_{1}\ne \varnothing$, and ${X}_{1}\setminus {cl}_{X}{X}_{0}$ is almost-$\omega$-resolvable, then ${C}_{\square }\left(X\right)$ is homeomorphic to a free topological sum of $\le |\stackrel{^}{C}\left({X}_{1}\right)|$ copies of $\square {ℝ}^{{X}_{0}}$, and, in this case, ${C}_{\square }\left(X\right)\cong \square {ℝ}^{{X}_{0}}$ if and only if $|\stackrel{^}{C}\left({X}_{1}\right)|\le {2}^{|{X}_{0}|}$. We conclude that for a space $X$ such that ${X}_{0}$ is ${F}_{\sigma }$, ${C}_{\square }\left(X\right)$ is never normal if $|{X}_{0}|>{\aleph }_{0}$ [La], and, assuming CH, ${C}_{\square }\left(X\right)$ is paracompact if $|{X}_{0}|={\aleph }_{0}$ [Ru2]. We also analyze ${C}_{\square }\left(X\right)$ when $|{X}_{1}|=1$ and when $X$ is countably compact, and we scrutinize under what conditions $\square {ℝ}^{\kappa }$ is homeomorphic to some of its “$\Sigma$-products"; in particular, we prove that $\square {ℝ}^{\omega }$ is homeomorphic to each of its subspaces $\left\{f\in \square {ℝ}^{\omega }:\left\{n\in \omega :f\left(n\right)=0\right\}\in p\right\}$ for every $p\in {\omega }^{*}$, and it is homeomorphic to $\left\{f\in \square {ℝ}^{\omega }:\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ϵ>0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left\{n\in \omega :|f\left(n\right)|<ϵ\right\}\in {ℱ}_{0}\right\}$ where ${ℱ}_{0}$ is the Fréchet filter on $\omega$.

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