| Title:
|
Spaces of continuous functions, $\Sigma$-products and Box Topology (English) |
| Author:
|
Angoa, J. |
| Author:
|
Tamariz-Mascarúa, Á. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
69-94 |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a Tychonoff space $X$, we will denote by $X_0$ the set of its isolated points and $X_{1}$ will be equal to $X\setminus X_{0}$. The symbol $C(X)$ denotes the space of real-valued continuous functions defined on $X$. $\square\Bbb{R}^{\kappa}$ is the Cartesian product $\Bbb{R}^{\kappa}$ with its box topology, and $C_{\square}(X)$ is $C(X)$ with the topology inherited from $\square\Bbb{R}^{X}$. By $\widehat{C}(X_1)$ we denote the set $\{f\in C(X_1) : f$ can be continuously extended to all of $X\}$. A space $X$ is almost-$\omega$-resolvable if it can be partitioned by a countable family of subsets in such a way that every non-empty open subset of $X$ has a non-empty intersection with the elements of an infinite subcollection of the given partition. We analyze $C_\square (X)$ when $X_0$ is $F_\sigma$ and prove: (1) for every topological space $X$, if $X_{0}$ is $F_{\sigma}$ in $X$, and $\emptyset \ne X_{1}\subset \operatorname{cl}_{X}X_{0}$, then $C_{\square}(X)\cong \square\Bbb{R}^{X_{0}}$; (2) for every space $X$ such that $X_{0}$ is $F_{\sigma}$, $\operatorname{cl}_{X}X_{0}\cap X_{1}\ne \emptyset$, and $X_1 \setminus \operatorname{cl}_X X_0$ is almost-$\omega$-resolvable, then $C_{\square}(X)$ is homeomorphic to a free topological sum of $\leq |\widehat{C}(X_1)|$ copies of $\square\Bbb{R}^{X_{0}}$, and, in this case, $C_{\square}(X) \cong \square\Bbb{R}^{X_{0}}$ if and only if $|\widehat{C}(X_1)|\leq 2^{|X_{0}|}$. We conclude that for a space $X$ such that $X_0$ is $F_\sigma$, $C_\square(X)$ is never normal if $|X_0| >\aleph _0$ [La], and, assuming CH, $C_\square (X)$ is paracompact if $|X_0| = \aleph _0$ [Ru2]. We also analyze $C_\square(X)$ when $|X_1| = 1$ and when $X$ is countably compact, and we scrutinize under what conditions $\square\Bbb{R}^\kappa$ is homeomorphic to some of its ``$\Sigma$-products"; in particular, we prove that $\square\Bbb{R}^\omega$ is homeomorphic to each of its subspaces $\{f \in \square\Bbb{R}^\omega : \{n\in \omega : f(n) = 0\}\in p\}$ for every $p \in \omega^*$, and it is homeomorphic to $\{f \in \square\Bbb{R}^\omega : \,\, \forall \,\, \epsilon > 0 \,\, \{n\in \omega : |f(n)| < \epsilon\} \in {\Cal{F}}_0\}$ where $\Cal F_0$ is the Fréchet filter on $\omega$. (English) |
| Keyword:
|
spaces of real-valued continuous functions |
| Keyword:
|
box topology |
| Keyword:
|
$\Sigma$-product |
| Keyword:
|
almost-$\omega$-resolvable space |
| MSC:
|
54B10 |
| MSC:
|
54C35 |
| MSC:
|
54D15 |
| idZBL:
|
Zbl 1150.54015 |
| idMR:
|
MR2223968 |
| . |
| Date available:
|
2009-05-05T16:55:40Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119575 |
| . |
| Reference:
|
[Ar] Arkhangel'skii A.V.: Topological Function Spaces.Mathematics and its Applications, vol. 78, Kluwer Academic Publishers Dordrecht, Boston, London (1992). MR 1144519 |
| Reference:
|
[BNS] Beckenstein E., Narici L., Suffel C.: Topological Algebras.North Holland Mathematics Studies, vol. 24, North Holland Amsterdam, New York, Oxford (1977). Zbl 0348.46041, MR 0473835 |
| Reference:
|
[CH] Comfort W.W., Hager W.: Estimates for the number of real-valued continuous functions.Trans. Amer. Math. Soc. 150 (1970), 619-631. Zbl 0199.57504, MR 0263016 |
| Reference:
|
[DH] Di Malo G., Holá L'.: Recent Progress in Function Spaces.Seconda Università degli Studi di Napoli, Quaderni di Matematica 3 (1998). MR 1762348 |
| Reference:
|
[vD] van Douwen E.K.: The box product of countably many metrizable spaces need not be normal.Fund. Math. 88 (1975), 127-132. MR 0385781 |
| Reference:
|
[E] Engelking R.: General Topology.Heldermann Verlag Berlin (1989). Zbl 0684.54001, MR 1039321 |
| Reference:
|
[GJ] Gillman L., Jerison M.: Rings of Continuous Functions.Graduate Texts in Mathematics, Springer New York, Heidelberg, Berlin (1976). Zbl 0327.46040, MR 0407579 |
| Reference:
|
[H] Hodel R.: Cardinal Functions I.Handbook of Set Theoretic Topology, (K. Kunen, J. Vaughan, Eds.), North Holland Amsterdam, New York, Oxford, Tokyo (1984, pp.1-61). MR 0776620 |
| Reference:
|
[Kn] Knight C.J.: Box topologies.Quart. J. Math. 15 (1964), 41-54. Zbl 0122.17404, MR 0160184 |
| Reference:
|
[Ku] Kunen K.: On paracompactness of box products of compact spaces.Trans. Amer. Math. Soc. 240 (1978), 307-316. MR 0514975 |
| Reference:
|
[KST] Kunen K., Szymansky A., Tall F.: Baire irresolvable spaces and ideal theory.Ann. Math. Silesiana 14 (1986), 98-107. MR 0861505 |
| Reference:
|
[La] Lawrence L.B.: Failure of normality in the box product of uncountably many real lines.Trans. Amer. Math. Soc. 348 (1996), 187-203. Zbl 0864.54017, MR 1303123 |
| Reference:
|
[NyP] Nyikos P., Piatkiewicz L.: Paracompact subspaces in the Box product Topology.Proc. Amer. Math. Soc. 124 (1996), 303-314. MR 1327033 |
| Reference:
|
[Ru1] Rudin M.E.: A normal space $X$ for which $X \times I$ is not normal.Fund. Math. 73 (1971), 179-186. Zbl 0224.54019, MR 0293583 |
| Reference:
|
[Ru2] Rudin M.E.: The box product of countably many compact metric spaces.General Topology Appl. 2 (1972), 293-298. Zbl 0243.54015, MR 0324619 |
| Reference:
|
[Ru3] Rudin M.E.: Lectures on set theoretic topology.Conference Board of the Mathematical Sciencie, Amer. Math. Soc. (1975). Zbl 0318.54001, MR 0367886 |
| Reference:
|
[TV] Tamariz-Mascarúa A., Villegas-Rodríguez H.: Spaces of continuous functions, box products and almost-$ømega$-resolvable spaces.Comment. Math. Univ. Carolinae 43 2 (2002), 687-705. Zbl 1090.54011, MR 2045790 |
| Reference:
|
[Ti] Tietze H.: Beitrage zur allgemeinen Topologie I.Math. Ann. 88 (1923), 280-312. MR 1512131 |
| Reference:
|
[V] Vaughan J.E.: Non-normal products of $ømega _{\mu}$metrizable spaces.Proc. Amer. Math. Soc. 51 (1975), 203-208. MR 0370464 |
| Reference:
|
[Wi] Williams S.W.: Box products.Handbook of Set-Theoretic Topology (K. Kunen, J. Vaughan, Eds.), North Holland Amsterdam, New York, Oxford, Tokyo (1984), 169-200. Zbl 0568.54011, MR 0776623 |
| . |
