Wolf Iberkleid, Ramiro Lafuente-Rodriguez, Warren Wm. McGovern
The regular topology on $C(X)$

Comment.Math.Univ.Carolin. 52,3 (2011) 445-461.

Abstract:Hewitt [{\it Rings of real-valued continuous functions. I.\/}, Trans. Amer. Math. Soc. {\bf 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 [{\it Nonnormality of spaces of real functions\/}, Topology Appl. {\bf 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, {\it The $m$-topology on $C_m(X)$ revisited\/}, Topology Appl. {\bf 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.

Keywords: DRS-space, Stone-\v Cech compactification, rings of continuous functions, $C(X)$
AMS Subject Classification: 54C35 54G99

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