| Title:
|
$G_\delta$-modification of compacta and cardinal invariants (English) |
| Author:
|
Arhangel'skii, A. V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
95-101 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Given a space $X$, its $G_\delta $-subsets form a basis of a new space $X_\omega $, called the $G_\delta $-modification of $X$. We study how the assumption that the $G_\delta $-modification $X_\omega $ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_\omega $ is discrete and, hence, homogeneous. Thus, $X_\omega $ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_\delta $-modification of $X$ is homogeneous, then the weight $w(X)$ of $X$ does not exceed $2^\omega $ (Theorem 1). We also establish that if a compact Hausdorff space of countable tightness is covered by a family of $G_\delta $-subspaces of the weight $\leq c=2^\omega $, then the weight of $X$ is not greater than $2^\omega $ (Theorem 4). Several other related results are obtained, a few new open questions are formulated. Fedorchuk's hereditarily separable compactum of the cardinality greater than $c=2^\omega $ is shown to be $G_\delta $-homogeneous under CH. Of course, it is not homogeneous when given its own topology. (English) |
| Keyword:
|
weight |
| Keyword:
|
tightness |
| Keyword:
|
$G_\delta $-modification |
| Keyword:
|
character |
| Keyword:
|
Lindelöf degree |
| Keyword:
|
homogeneous space |
| MSC:
|
54A25 |
| MSC:
|
54B10 |
| idZBL:
|
Zbl 1150.54004 |
| idMR:
|
MR2223969 |
| . |
| Date available:
|
2009-05-05T16:55:45Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119576 |
| . |
| Reference:
|
[1] Arhangel'skii A.V.: On cardinal invariants.In: General Topology and its Relations to Modern Analysis and Algebra 3. Proceedings of the Third Prague Topological Symposium, 1971, 37-46. Academia Publishing House, Czechoslovak Academy of Sciences, Prague, 1972. MR 0410629 |
| Reference:
|
[2] Arhangel'skii A.V.: Structure and classification of topological spaces and cardinal invariants.Russian Math. Surveys 33 (1978), 33-96. MR 0526012 |
| Reference:
|
[3] Arhangel'skii A.V.: Topological homogeneity, topological groups and their continuous images.Russian Math. Surveys 42 (1987), 83-131. MR 0898622 |
| Reference:
|
[4] Arhangel'skii A.V.: Homogeneity of powers of spaces and the character.Proc. Amer. Math. Soc. 133 (2005), 2165-2172. Zbl 1068.54005, MR 2137884 |
| Reference:
|
[5] Arhangel'skii A.V.: A weak algebraic structure on topological spaces and cardinal invariants.Topology Proc. 28 1 (2004), 1-18. MR 2105445 |
| Reference:
|
[6] Arhangel'skii A.V., van Mill J., Ridderbos G.J.: A new bound on the cardinality of power homogeneous compacta.Houston J. Math., to appear. |
| Reference:
|
[7] Balogh Z.: On compact Hausdorff spaces of countable tightness.Proc. Amer. Math. Soc. 105 (1989), 755-764. Zbl 0687.54006, MR 0930252 |
| Reference:
|
[8] Dow A.: An introduction to applications of elementary submodels to topology.Topology Proc. 13 (1988), 17-72. Zbl 0696.03024, MR 1031969 |
| Reference:
|
[9] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[10] Fedorchuk V.: On the cardinality of hereditarily separable compact Hausdorff spaces.Soviet Math. Dokl. 16 (1975), 651-655. Zbl 0331.54029 |
| Reference:
|
[11] Kunen K.: Box products of compact spaces.Trans. Amer. Math. Soc. 240 (1978), 307-316. Zbl 0386.54003, MR 0514975 |
| Reference:
|
[12] Levy R., Rice M.D.: Normal $P$-spaces and the $G_\delta $-topology Colloq. Math..44 (1981), 227-240. MR 0652582 |
| Reference:
|
[13] Meyer P.R.: The Baire order problem for compact spaces.Duke Math. J. 33 (1966), 33-40. Zbl 0138.17602, MR 0190897 |
| Reference:
|
[14] Pytkeev E.G.: About the $G_\lambda $-topology and the power of some families of subsets on compacta.Colloq. Math. Soc. Janos Bolyai, 41. Topology and Applications, Eger (Hungary), 1983, pp.517-522. MR 0863935 |
| Reference:
|
[15] de la Vega R.: A new bound on the cardinality of homogeneous compacta.Topology Appl., to appear. Zbl 1098.54002, MR 2239075 |
| . |
