| Title:
|
On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces (English) |
| Author:
|
Okunev, Oleg |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
629-635 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p(X)$ is Lindelöf, $Y=X\cup\{p\}$, and the point $p$ has countable character in $Y$, then $C_p(Y)$ is Lindelöf; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l(C_p(Y)^\omega)\le l(C_p(X)^\omega)$ and $\operatorname{ext}(C_p(Y)^\omega)\le \operatorname{ext}(C_p(X)^\omega)$. (English) |
| Keyword:
|
pointwise convergence |
| Keyword:
|
Lindelöf property |
| MSC:
|
54C35 |
| MSC:
|
54D20 |
| idZBL:
|
Zbl 1212.54052 |
| idMR:
|
MR2583139 |
| . |
| Date available:
|
2009-12-22T10:05:06Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137452 |
| . |
| Reference:
|
[Ar1] Arhangel'skii A.V.: On linear topological and topological classification of spaces $C_p(X)$.Zb. Rad. 3 (1989), 3--12. MR 1017630 |
| Reference:
|
[Arh2] Arhangel'skii A.V.: Topological Function Spaces.Kluwer Acad. Publ. Dordrecht (1992). MR 1485266 |
| Reference:
|
[Buz] Buzyakova R.Z.: How sensitive is $C_p(X,Y)$ to changes in $X$ and/or $Y$?.Comment. Math. Univ. Carolin. 49 4 (2008), 657--665. Zbl 1212.54051, MR 2493945 |
| Reference:
|
[Eng] Engelking R.: General Topology.Sigma Series in Pure Mathematics, 6, Helderman, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[KOS] Kubiš W., Okunev O., Szeptycki P.J.: On some classes of Lindelöf$\Sigma$-spaces.Topology Appl. 153 (2006), 2574--2590. Zbl 1102.54028, MR 2243735, 10.1016/j.topol.2005.09.009 |
| Reference:
|
[Oku] Okunev O.: On Lindelöf $\Sigma$-spaces of continuous functions in the pointwise topology.Topology Appl. 49 (1993), 149--166. Zbl 0796.54026, MR 1206222, 10.1016/0166-8641(93)90041-B |
| Reference:
|
[Pol] Pol R.: A theorem on the weak topology of $C(X)$ for compact scattered $X$.Fund. Math. 106 2 (1980), 135--140. Zbl 0444.54010, MR 0580591 |
| Reference:
|
[RJ] Rogers A., Jayne E. (Eds.): Analytic Sets.Academic Press, London, 1980. Zbl 0589.54047 |
| . |
