The regular topology on C(X)

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.