Mathematics Faculty Articles


The Regular Topology on C(X)

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Commentationes Mathematicae Universitatis Carolinae


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







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Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the m-topology on C(X), denoted Cm(X), and demonstrated that certain topological properties of X could be characterized by certain topological properties of Cm(X). For example, he showed that X is pseudocompact if and only if Cm(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 Cm(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 Cm(X) satisfied the property that all countable subsets of Cm(X) are closed. In J. Gomez-Perez and W.Wm. McGovern, The m-topology on Cm(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 Cr(X). In Section 2, we characterize when Cr(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.

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