# Article

Full entry | PDF   (0.3 MB)
Keywords:
DRS-space; Stone-Čech compactification; rings of continuous functions; $C(X)$
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.
References:
[1] Darnel M.: Theory of Lattice-Ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, Inc., New York, 1995. MR 1304052 | Zbl 0810.06016
[2] Di Maio G., Holá L., Holý D., McCoy R.A.: Topologies on the space of continuous functions. Topology Appl. 86 (1998), no. 2, 105–122. DOI 10.1016/S0166-8641(97)00114-4 | MR 1621396
[3] Engelking R.: General Topology. Polish Sci. Publishers, Berlin, 1977. MR 0500780 | Zbl 0684.54001
[4] Fine N.J., Gillman L.: Extension of continuous functions in $\beta \mathbb N$. Bull. Amer. Math. Soc. 66 (1960), 376-381. DOI 10.1090/S0002-9904-1960-10460-0 | MR 0123291
[5] Fine N.J., Gillman L., Lambek J.: Rings of Quotients of Rings of Functions. Lecture Note Series, McGill University Press, Montreal, 1966. MR 0200747 | Zbl 0143.35704
[6] Gillman L., Jerison M.: Rings of Continuous Functions. The University Series in Higher Mathematics, D. Van Nostrand, Princeton, 1960. MR 0116199 | Zbl 0327.46040
[7] Gomez-Perez J., McGovern W.Wm.: The $m$-topology on $C_m(X)$ revisited. Topology Appl. 153 (2006), no. 11, 1838–1848. DOI 10.1016/j.topol.2005.06.016 | MR 2227030 | Zbl 1117.54045
[8] Hager A., Martinez J.: Fraction-dense algebras and spaces. Canad. J. Math. 45 (1993), no. 5, 977–996. DOI 10.4153/CJM-1993-054-6 | MR 1239910 | Zbl 0795.06017
[9] Henriksen M., Woods R.G.: Cozero complemented space; when the space of minimal prime ideals of a $C(X)$ is compact. Topology Appl. 141 (2004), no. 1–3, 147–170. DOI 10.1016/j.topol.2003.12.004 | MR 2058685
[10] Hewitt E.: Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64 (1948), 45–99. DOI 10.1090/S0002-9947-1948-0026239-9 | MR 0026239 | Zbl 0032.28603
[11] Kopperman R., Pajoohesh H., Richmond T.: Topologies arising from metrics valued in abelian $\ell$-groups. Algebra Universalis(to appear). MR 2817554
[12] Levy R.: Almost $P$-spaces. Canad. J. Math. 29 (1977), no. 2, 284–288. DOI 10.4153/CJM-1977-030-7 | MR 0464203 | Zbl 0342.54032
[13] Levy R., Shapiro J.: Rings of quotients of rings of functions. Topology Appl. 146/146 (2005), 253–265. DOI 10.1016/j.topol.2003.03.003 | MR 2107150 | Zbl 1064.54032
[14] Porter J., Woods R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York, 1998. MR 0918341 | Zbl 0652.54016
[15] van Douwen E.: Nonnormality of spaces of real functions. Topology Appl. 39 (1991), 3–32. DOI 10.1016/0166-8641(91)90072-T | MR 1103988

Partner of