# Closure-preserving covers in function spaces

Commentationes Mathematicae Universitatis Carolinae (2010)

- Volume: 51, Issue: 4, page 693-703
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topSánchez, David Guerrero. "Closure-preserving covers in function spaces." Commentationes Mathematicae Universitatis Carolinae 51.4 (2010): 693-703. <http://eudml.org/doc/246906>.

@article{Sánchez2010,

abstract = {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.},

author = {Sánchez, David Guerrero},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {closure-preserving covers; function spaces; compact spaces; pointwise convergence topology; topological game; winning strategy; closure-preserving covers; function spaces; compact space; pointwise convergence topology; topological game; winning strategy},

language = {eng},

number = {4},

pages = {693-703},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Closure-preserving covers in function spaces},

url = {http://eudml.org/doc/246906},

volume = {51},

year = {2010},

}

TY - JOUR

AU - Sánchez, David Guerrero

TI - Closure-preserving covers in function spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2010

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 51

IS - 4

SP - 693

EP - 703

AB - 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.

LA - eng

KW - closure-preserving covers; function spaces; compact spaces; pointwise convergence topology; topological game; winning strategy; closure-preserving covers; function spaces; compact space; pointwise convergence topology; topological game; winning strategy

UR - http://eudml.org/doc/246906

ER -

## References

top- Arkhangelskii A.V., Topological function spaces, Mathematics and its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. MR1144519
- Junnila H.J.K., 10.1090/S0002-9947-1979-0525679-9, Trans. Amer. Math. Soc. 249 (1979), no. 2, 373–385. Zbl0404.54017MR0525679DOI10.1090/S0002-9947-1979-0525679-9
- Engelking R., General Topology, Heldermann, Berlin, 1989. Zbl0684.54001MR1039321
- Potoczny H.B., 10.1016/0016-660X(72)90015-3, General Topology and Appl. 3 (1973), 243–248. Zbl0263.54010MR0322805DOI10.1016/0016-660X(72)90015-3
- Potoczny H.B., Junnila H.J.K., 10.1090/S0002-9939-1975-0388337-1, Proc. Amer. Math. Soc. 53 (1975), no. 2, 523–529. Zbl0318.54018MR0388337DOI10.1090/S0002-9939-1975-0388337-1
- Rogers C.A., Jayne J.E., -analytic Sets, Rogers C.A., Jayne J.E., et al., “Analytic Sets", Academic Press, London, 1980, pp. 2–181.
- Shakmatov D.B., Tkachuk V.V., When is the space ${C}_{p}\left(X\right)$$\sigma $-countably compact?, Vestnik Moskov. Univ. Mat. 41 (1986), no. 1, 73–75.
- Telgársky R., Spaces defined by topological games, Fund. Math. 88 (1975), no. 3, 193–223. MR0380708
- Tkachuk V.V., 10.1016/0166-8641(86)90023-4, Topology Appl. 22 (1986), no. 3, 241–253. MR0842658DOI10.1016/0166-8641(86)90023-4
- Tkachuk V.V., The decomposition of ${C}_{p}\left(X\right)$ into a countable union of subspaces with “good” properties implies “good” properties of ${C}_{p}\left(X\right)$, Trans. Moscow Math. Soc. 55 (1994), 239–248. MR1468461

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.