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 $X$. $\square \Bbb {R}^{\kappa }$ 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 $\emptyset \not =X_{1}\subset cl_{X}X_{0}$, then $C_{\square }(X)\cong \square \Bbb {R}^{X_{0}}$; (2) for every space $X$ such that $X_{0}$ is $F_{\sigma }$, $cl_{X}X_{0}\cap X_{1}\not =\emptyset $, and $X_1 \setminus cl_X X_0$ 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 Fr\'echet filter on $\omega $.
Keywords: spaces of real-valued continuous functions, box topology, $\Sigma $-product, almost-$\omega $-resolvable space
AMS Subject Classification: 54C35, 54B10, 54D15