| Title:
|
Connectedness of some rings of quotients of $C(X)$ with the $m$-topology (English) |
| Author:
|
Azarpanah, F. |
| Author:
|
Paimann, M. |
| Author:
|
Salehi, A. R. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
63-76 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article we define the $m$-topology on some rings of quotients of $C(X)$. Using this, we equip the classical ring of quotients $q(X)$ of $C(X)$ with the $m$-topology and we show that $C(X)$ with the $r$-topology is in fact a subspace of $q(X)$ with the $m$-topology. Characterization of the components of rings of quotients of $C(X)$ is given and using this, it turns out that $q(X)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C(X)$ with $r$-topology is connected. We also observe that the maximal ring of quotients $Q(X)$ of $C(X)$ with the $m$-topology is connected if and only if $X$ is finite. Finally for each point $x$, we introduce a natural ring of quotients of $C(X)/O_x$ which is connected with the $m$-topology. (English) |
| Keyword:
|
$r$-topology |
| Keyword:
|
$m$-topology |
| Keyword:
|
almost $P$-space |
| Keyword:
|
pseudocompact space |
| Keyword:
|
component |
| Keyword:
|
classical ring of quotients of $C(X)$ |
| MSC:
|
54C35 |
| MSC:
|
54C40 |
| idZBL:
|
Zbl 06433806 |
| idMR:
|
MR3311578 |
| DOI:
|
10.14712/1213-7243.015.106 |
| . |
| Date available:
|
2015-03-10T17:37:14Z |
| Last updated:
|
2017-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144189 |
| . |
| Reference:
|
[1] Azarpanah F., Manshoor F., Mohamadian R.: Connectedness and compactness in $C(X)$ with the $m$-topology and generalized $m$-topology.Topology Appl. 159 (2012), 3486–3493. Zbl 1255.54009, MR 2964861, 10.1016/j.topol.2012.08.010 |
| Reference:
|
[2] Comfort W.W., Negrepontis S.: The Theory of Ultrafilters.Die Grundelehren der Math. Wissenschaften, 211, Springer, New York-Heidelberg-Berlin, 1974. Zbl 0298.02004, MR 0396267 |
| Reference:
|
[3] Dashiell F., Hager A., Henriksen M.: Order-Cauchy completions of rings and vector lattices of continuous functions.Canad. J. Math. 32 (1980), no. 3, 657–685. Zbl 0462.54009, MR 0586984, 10.4153/CJM-1980-052-0 |
| Reference:
|
[4] Engelking R.: General Topology.PWN-Polish Sci. Publ., Warsaw, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[5] Fine N.J., Gillman L., Lambek J.: Rings of quotients of rings of continuous functions.Lecture Note Series, McGill University Press, Montreal, 1966. MR 0200747 |
| Reference:
|
[6] Gillman L., Jerison M.: Rings of Continuous Functions.Springer, New York-Heidelberg, 1976. Zbl 0327.46040, MR 0407579 |
| Reference:
|
[7] Gomez-Pérez J., McGovern W.W.: The $m$-topology on $C_m(X)$ revisited.Topology Appl. 153 (2006), 1838–1848. Zbl 1117.54045, MR 2227030, 10.1016/j.topol.2005.06.016 |
| Reference:
|
[8] Iberkleid W., Lafuente-Rodriguez R., McGovern W.Wm.: The regular topology on $C(X)$.Comment. Math. Univ. Carolin. 52 (2011), no. 3, 445–461. Zbl 1249.54037, MR 2843236 |
| Reference:
|
[9] Henriksen M., Mitra B.: $C(X)$ can sometimes determine $X$ without $X$ being realcompact.Comment. Math. Univ. Carolin. 46 (2005), no. 4, 711–720. Zbl 1121.54035, MR 2259501 |
| Reference:
|
[10] Hewitt E.: Rings of real-valued continuous functions I.Trans. Amer. Math. Soc. 48 (64) (1948), 54–99. Zbl 0032.28603, MR 0026239 |
| Reference:
|
[11] Johnson D.G., Mandelker M.: Functions with pseudocompact support.General Topology Appl. 3 (1973), 331–338. Zbl 0277.54009, MR 0331310, 10.1016/0016-660X(73)90020-2 |
| Reference:
|
[12] Di Maio G., Holá L'., Holý D., McCoy D.: Topology on the space of continuous functions.Topology Appl. 86 (1998), 105–122. MR 1621396, 10.1016/S0166-8641(97)00114-4 |
| Reference:
|
[13] Swardson M.: A generalization of $F$-spaces and some topological characterizations of GCH.Trans. Amer. Math. Soc. 279 (1983), 661–675. Zbl 0535.54023, MR 0709575 |
| Reference:
|
[14] van Douwen E.K.: Nonnormality or hereditary paracompactness of some spaces of real functions.Topology Appl. 39 (1991), 3–32. Zbl 0755.54008, MR 1103988, 10.1016/0166-8641(91)90072-T |
| Reference:
|
[15] Walker R.C.: The Stone-Čech Compactification.Springer, Berlin-Heidelberg-New York, 1974. Zbl 0292.54001, MR 0380698 |
| Reference:
|
[16] Warner S.: Topological Rings.North-Holland Mathematics Studies, 178, North-Holland, Ansterdam, 1993. Zbl 0785.13008, MR 1240057 |
| Reference:
|
[17] Wilard S.: General Topology.Addison-Wesley, Readings, MA, 1970. MR 0264581 |
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