| Title:
|
Closure-preserving covers in function spaces (English) |
| Author:
|
Sánchez, David Guerrero |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
693-703 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is shown that if $C_p(X)$ admits a closure-preserving cover by closed $\sigma$-compact sets then $X$ is finite. If $X$ is compact and $C_p(X)$ has a closure-preserving cover by separable subspaces then $X$ is metrizable. We also prove that if $C_p(X,[0,1])$ has a closure-preserving cover by compact sets, then $X$ is discrete. (English) |
| Keyword:
|
closure-preserving covers |
| Keyword:
|
function spaces |
| Keyword:
|
compact spaces |
| Keyword:
|
pointwise convergence topology |
| Keyword:
|
topological game |
| Keyword:
|
winning strategy |
| MSC:
|
54C35 |
| idZBL:
|
Zbl 1224.54045 |
| idMR:
|
MR2858270 |
| . |
| Date available:
|
2010-11-30T16:29:32Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140847 |
| . |
| Reference:
|
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| Reference:
|
[2] Junnila H.J.K.: Metacompactness, paracompactness, and interior-preserving open covers.Trans. Amer. Math. Soc. 249 (1979), no. 2, 373–385. Zbl 0404.54017, MR 0525679, 10.1090/S0002-9947-1979-0525679-9 |
| Reference:
|
[3] Engelking R.: General Topology.Heldermann, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[4] Potoczny H.B.: Closure-preserving families of compact sets.General Topology and Appl. 3 (1973), 243–248. Zbl 0263.54010, MR 0322805, 10.1016/0016-660X(72)90015-3 |
| Reference:
|
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| Reference:
|
[6] Rogers C.A., Jayne J.E., $K$-analytic Sets, Rogers C.A., Jayne J.E.: et al..“Analytic Sets", Academic Press, London, 1980, pp. 2–181. |
| Reference:
|
[7] Shakmatov D.B., Tkachuk V.V.: When is the space $C_p(X)$ $\sigma$-countably compact?.Vestnik Moskov. Univ. Mat. 41 (1986), no. 1, 73–75. |
| Reference:
|
[8] Telgársky R.: Spaces defined by topological games.Fund. Math. 88 (1975), no. 3, 193–223. MR 0380708 |
| Reference:
|
[9] Tkachuk V.V.: The spaces $C_p(X)$: decomposition into a countable union of bounded subspaces and completeness properties.Topology Appl. 22 (1986), no. 3, 241–253. MR 0842658, 10.1016/0166-8641(86)90023-4 |
| Reference:
|
[10] Tkachuk V.V.: The decomposition of $ C_p(X)$ into a countable union of subspaces with “good” properties implies “good” properties of $ C_p(X)$.Trans. Moscow Math. Soc. 55 (1994), 239–248. MR 1468461 |
| . |
