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Keywords:
spaces of real-valued continuous functions; box topology; $\Sigma$-product; almost-$\omega$-resolvable space
spaces of real-valued continuous functions; box topology; $\Sigma$-product; almost-$\omega$-resolvable space
Summary:
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 $X$. $\square\Bbb{R}^{\kappa}$ is the Cartesian product $\Bbb{R}^{\kappa}$ with its box topology, and $C_{\square}(X)$ is $C(X)$ with the topology inherited from $\square\Bbb{R}^{X}$. By $\widehat{C}(X_1)$ we denote the set $\{f\in C(X_1) : f$ can be continuously extended to all of $X\}$. 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 $C_{\square}(X)\cong \square\Bbb{R}^{X_{0}}$; (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 $\leq |\widehat{C}(X_1)|$ copies of $\square\Bbb{R}^{X_{0}}$, and, in this case, $C_{\square}(X) \cong \square\Bbb{R}^{X_{0}}$ if and only if $|\widehat{C}(X_1)|\leq 2^{|X_{0}|}$. 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 $\square\Bbb{R}^\kappa$ is homeomorphic to some of its ``$\Sigma$-products"; in particular, we prove that $\square\Bbb{R}^\omega$ is homeomorphic to each of its subspaces $\{f \in \square\Bbb{R}^\omega : \{n\in \omega : f(n) = 0\}\in p\}$ for every $p \in \omega^*$, and it is homeomorphic to $\{f \in \square\Bbb{R}^\omega : \,\, \forall \,\, \epsilon > 0 \,\, \{n\in \omega : |f(n)| < \epsilon\} \in {\Cal{F}}_0\}$ where $\Cal F_0$ is the Fréchet filter on $\omega$.
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