The regular topology on C ( X )

Wolf Iberkleid; Ramiro Lafuente-Rodriguez; Warren Wm. McGovern

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 3, page 445-461
  • ISSN: 0010-2628

Abstract

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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 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 β 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.

How to cite

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Iberkleid, Wolf, Lafuente-Rodriguez, Ramiro, and McGovern, Warren Wm.. "The regular topology on $C(X)$." Commentationes Mathematicae Universitatis Carolinae 52.3 (2011): 445-461. <http://eudml.org/doc/247067>.

@article{Iberkleid2011,
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 $\beta X$ 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.},
author = {Iberkleid, Wolf, Lafuente-Rodriguez, Ramiro, McGovern, Warren Wm.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {DRS-space; Stone-Čech compactification; rings of continuous functions; $C(X)$; ; -topology; -topology; DRS-space; weak -space; cardinal invariants},
language = {eng},
number = {3},
pages = {445-461},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The regular topology on $C(X)$},
url = {http://eudml.org/doc/247067},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Iberkleid, Wolf
AU - Lafuente-Rodriguez, Ramiro
AU - McGovern, Warren Wm.
TI - The regular topology on $C(X)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 3
SP - 445
EP - 461
AB - 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.
LA - eng
KW - DRS-space; Stone-Čech compactification; rings of continuous functions; $C(X)$; ; -topology; -topology; DRS-space; weak -space; cardinal invariants
UR - http://eudml.org/doc/247067
ER -

References

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  1. Darnel M., Theory of Lattice-Ordered Groups, Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, Inc., New York, 1995. Zbl0810.06016MR1304052
  2. Di Maio G., Holá L., Holý D., McCoy R.A., 10.1016/S0166-8641(97)00114-4, Topology Appl. 86 (1998), no. 2, 105–122. MR1621396DOI10.1016/S0166-8641(97)00114-4
  3. Engelking R., General Topology, Polish Sci. Publishers, Berlin, 1977. Zbl0684.54001MR0500780
  4. Fine N.J., Gillman L., 10.1090/S0002-9904-1960-10460-0, Bull. Amer. Math. Soc. 66 (1960), 376-381. MR0123291DOI10.1090/S0002-9904-1960-10460-0
  5. Fine N.J., Gillman L., Lambek J., Rings of Quotients of Rings of Functions, Lecture Note Series, McGill University Press, Montreal, 1966. Zbl0143.35704MR0200747
  6. Gillman L., Jerison M., Rings of Continuous Functions, The University Series in Higher Mathematics, D. Van Nostrand, Princeton, 1960. Zbl0327.46040MR0116199
  7. Gomez-Perez J., McGovern W.Wm., 10.1016/j.topol.2005.06.016, Topology Appl. 153 (2006), no. 11, 1838–1848. Zbl1117.54045MR2227030DOI10.1016/j.topol.2005.06.016
  8. Hager A., Martinez J., 10.4153/CJM-1993-054-6, Canad. J. Math. 45 (1993), no. 5, 977–996. Zbl0795.06017MR1239910DOI10.4153/CJM-1993-054-6
  9. Henriksen M., Woods R.G., 10.1016/j.topol.2003.12.004, Topology Appl. 141 (2004), no. 1–3, 147–170. MR2058685DOI10.1016/j.topol.2003.12.004
  10. Hewitt E., 10.1090/S0002-9947-1948-0026239-9, Trans. Amer. Math. Soc. 64 (1948), 45–99. Zbl0032.28603MR0026239DOI10.1090/S0002-9947-1948-0026239-9
  11. Kopperman R., Pajoohesh H., Richmond T., Topologies arising from metrics valued in abelian ℓ -groups, Algebra Universalis(to appear). MR2817554
  12. Levy R., 10.4153/CJM-1977-030-7, Canad. J. Math. 29 (1977), no. 2, 284–288. Zbl0342.54032MR0464203DOI10.4153/CJM-1977-030-7
  13. Levy R., Shapiro J., 10.1016/j.topol.2003.03.003, Topology Appl. 146/146 (2005), 253–265. Zbl1064.54032MR2107150DOI10.1016/j.topol.2003.03.003
  14. Porter J., Woods R.G., Extensions and Absolutes of Hausdorff Spaces, Springer, New York, 1998. Zbl0652.54016MR0918341
  15. van Douwen E., 10.1016/0166-8641(91)90072-T, Topology Appl. 39 (1991), 3–32. MR1103988DOI10.1016/0166-8641(91)90072-T

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