| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2011-08-15T19:24:43Z |
| Last updated:
|
2013-10-14 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141615 |
| . |
| Reference:
|
[1] Darnel M.: Theory of Lattice-Ordered Groups.Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, Inc., New York, 1995. Zbl 0810.06016, MR 1304052 |
| Reference:
|
[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. MR 1621396, 10.1016/S0166-8641(97)00114-4 |
| Reference:
|
[3] Engelking R.: General Topology.Polish Sci. Publishers, Berlin, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[4] Fine N.J., Gillman L.: Extension of continuous functions in $\beta \mathbb N$.Bull. Amer. Math. Soc. 66 (1960), 376-381. MR 0123291, 10.1090/S0002-9904-1960-10460-0 |
| Reference:
|
[5] Fine N.J., Gillman L., Lambek J.: Rings of Quotients of Rings of Functions.Lecture Note Series, McGill University Press, Montreal, 1966. Zbl 0143.35704, MR 0200747 |
| Reference:
|
[6] Gillman L., Jerison M.: Rings of Continuous Functions.The University Series in Higher Mathematics, D. Van Nostrand, Princeton, 1960. Zbl 0327.46040, MR 0116199 |
| Reference:
|
[7] Gomez-Perez J., McGovern W.Wm.: The $m$-topology on $C_m(X)$ revisited.Topology Appl. 153 (2006), no. 11, 1838–1848. Zbl 1117.54045, MR 2227030, 10.1016/j.topol.2005.06.016 |
| Reference:
|
[8] Hager A., Martinez J.: Fraction-dense algebras and spaces.Canad. J. Math. 45 (1993), no. 5, 977–996. Zbl 0795.06017, MR 1239910, 10.4153/CJM-1993-054-6 |
| Reference:
|
[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. MR 2058685, 10.1016/j.topol.2003.12.004 |
| Reference:
|
[10] Hewitt E.: Rings of real-valued continuous functions. I..Trans. Amer. Math. Soc. 64 (1948), 45–99. Zbl 0032.28603, MR 0026239, 10.1090/S0002-9947-1948-0026239-9 |
| Reference:
|
[11] Kopperman R., Pajoohesh H., Richmond T.: Topologies arising from metrics valued in abelian $\ell$-groups.Algebra Universalis(to appear). MR 2817554 |
| Reference:
|
[12] Levy R.: Almost $P$-spaces.Canad. J. Math. 29 (1977), no. 2, 284–288. Zbl 0342.54032, MR 0464203, 10.4153/CJM-1977-030-7 |
| Reference:
|
[13] Levy R., Shapiro J.: Rings of quotients of rings of functions.Topology Appl. 146/146 (2005), 253–265. Zbl 1064.54032, MR 2107150, 10.1016/j.topol.2003.03.003 |
| Reference:
|
[14] Porter J., Woods R.G.: Extensions and Absolutes of Hausdorff Spaces.Springer, New York, 1998. Zbl 0652.54016, MR 0918341 |
| Reference:
|
[15] van Douwen E.: Nonnormality of spaces of real functions.Topology Appl. 39 (1991), 3–32. MR 1103988, 10.1016/0166-8641(91)90072-T |
| . |
