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Keywords:
DRS-space; Stone-Čech compactification; rings of continuous functions; $C(X)$
DRS-space; Stone-Čech compactification; rings of continuous functions; $C(X)$
Summary:
Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45--99] defined the $m$-topology on $C(X)$, denoted $C_m(X)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m(X)$. For example, he showed that $X$ is pseudocompact if and only if $C_m(X)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with the right kind of space $X$, for $C_m(X)$ to be highly
non-metrizable. E. van Douwen [Nonnormality of spaces of real functions, Topology Appl. 39 (1991),
3--32] defined the class of DRS-spaces and showed that if $X$ was such a space, then $C_m(X)$ satisfied the property that all countable subsets of $C_m(X)$ are closed. In J. Gomez-Perez and W.Wm. McGovern, The $m$-topology on $C_m(X)$ revisited, Topology Appl. 153, (2006), no. 11, 1838--1848, the authors demonstrated the converse, completing the characterization. In this article we define a finer topology on $C(X)$ based on positive regular elements. It is the authors' opinion that the new topology is a more well-behaved topology with regards to passing from $C(X)$ to $C^*(X)$. In the first section we compute some common cardinal invariants of the preceding space $C_r(X)$. In Section 2, we characterize when $C_r(X)$ satisfies the property that all countable subsets are closed. We call such a space for which this happens a weak DRS-space and demonstrate that $X$ is a weak DRS-space if and only if $\beta X$ is a weak DRS-space. This is somewhat surprising as a DRS-space cannot be compact. In the third section we give an internal characterization of separable weak DRS-spaces and use this to show that a metrizable space is a weak DRS-space precisely when it is nowhere separable.
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