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Title: The regular topology on $C(X)$ (English)
Author: Iberkleid, Wolf
Author: Lafuente-Rodriguez, Ramiro
Author: McGovern, Warren Wm.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 52
Issue: 3
Year: 2011
Pages: 445-461
Summary lang: English
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Category: math
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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. (English)
Keyword: DRS-space
Keyword: Stone-Čech compactification
Keyword: rings of continuous functions
Keyword: $C(X)$
MSC: 54C35
MSC: 54G99
idZBL: Zbl 1249.54037
idMR: MR2843236
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Date available: 2011-08-15T19:24:43Z
Last updated: 2013-10-14
Stable URL: http://hdl.handle.net/10338.dmlcz/141615
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