## Mathematics Faculty Articles

#### Title

The Regular Topology on C(X)

#### Document Type

Article

#### Publication Date

1-1-2011

#### Publication Title

Commentationes Mathematicae Universitatis Carolinae

#### Keywords

DRS-Space, Stone-Cech compactification, Rings of continuous functions, C(X), r-topology, m-topology, DRS-Space, Weak P-space, Cardinal invariants

#### ISSN

0010-2628

#### Volume

52

#### Issue/No.

3

#### First Page

445

#### Last Page

461

#### Abstract

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 *βX* 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.

#### NSUWorks Citation

Iberkleid, Wolf; Lafuente-Rodriguez, Ramiro; and McGovern, Warren William, "The Regular Topology on C(X)" (2011). *Mathematics Faculty Articles*. 123.

https://nsuworks.nova.edu/math_facarticles/123