| Title:
|
Diagonals and discrete subsets of squares (English) |
| Author:
|
Burke, Dennis |
| Author:
|
Tkachuk, Vladimir V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
69-82 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $|D|=d(X)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma$-space $X$ and hence $X^\omega $ is $d$-separable. We give an example of a countably compact space $X$ such that $X^\omega $ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta = \{(x,x): x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta $ is a retract of $\overline{D}$ and the projection of $D$ on the first coordinate is dense in $X$. As a consequence, some properties that are not discretely reflexive in $X$ become discretely reflexive in $X\times X$. In particular, if $X$ is compact and $\overline{D}$ is Corson (Eberlein) compact for any discrete $D\subset X\times X$ then $X$ itself is Corson (Eberlein). Besides, a Lindelöf $p$-space $X$ is zero-dimensional if and only if $\overline{D}$ is zero-dimensional for any discrete $D\subset X\times X$. Under CH, we give an example of a crowded countable space $X$ such that every discrete subset of $X\times X$ is closed. In particular, the diagonal of $X$ cannot be contained in the closure of a discrete subspace of $X\times X$. (English) |
| Keyword:
|
diagonal |
| Keyword:
|
discrete subspaces |
| Keyword:
|
$d$-separable space |
| Keyword:
|
discrete reflexivity |
| Keyword:
|
Lindelöf $p$-space |
| Keyword:
|
Lindelöf $\Sigma $-space |
| Keyword:
|
finite powers |
| Keyword:
|
Corson compact spaces |
| Keyword:
|
Eberlein compact spaces |
| Keyword:
|
countably compact spaces |
| MSC:
|
54C10 |
| MSC:
|
54C25 |
| MSC:
|
54D25 |
| MSC:
|
54H11 |
| idMR:
|
MR3038072 |
| . |
| Date available:
|
2013-02-21T14:04:34Z |
| Last updated:
|
2015-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143153 |
| . |
| Reference:
|
[ATW] Alas O., Tkachuk V.V., Wilson R.G.: Closures of discrete sets often reflect global properties.Topology Proc. 25 (2000), 27–44. Zbl 1002.54021, MR 1875581 |
| Reference:
|
[Ar] Arhangel'skii A.V.: A class of spaces which contains all metric and all locally compact spaces (in Russian).Mat. Sb. 67 (109) (1965), 1 55–88. MR 0190889 |
| Reference:
|
[BT] Burke D., Tkachuk V.V.: Discrete reflexivity and complements of the diagonal.Acta Math. Hungarica, to appear. |
| Reference:
|
[DTTW] Dow A., Tkachenko M.G., Tkachuk V.V., Wilson R.G.: Topologies generated by discrete subspaces.Glasnik Mat. 37(57) (2002), 189–212. Zbl 1009.54005, MR 1918105 |
| Reference:
|
[vD] van Douwen E.K.: Applications of maximal topologies.Topology Appl. 51 (1993), 125–139. Zbl 0845.54028, MR 1229708, 10.1016/0166-8641(93)90145-4 |
| Reference:
|
[En] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[Gr] Gruenhage G.: Generalized Metric Spaces.Handbook of Set-Theoretic Topology, Ed. by K. Kunen and J.E. Vaughan, Elsevier Science Publisher, New York, 1984, pp. 423–501. Zbl 0794.54034, MR 0776629 |
| Reference:
|
[JSz] Juhász I., Szentmiklossy Z.: On $d$-separability of powers and $C_p(X)$.Topology Appl. 155 (2008), 277–281. Zbl 1134.54002, MR 2380265 |
| Reference:
|
[Tk1] Tkachuk V.V.: Spaces that are projective with respect to classes of mappings.Trans. Moscow Math. Soc. 50 (1988), 139–156. Zbl 0662.54007, MR 0912056 |
| Reference:
|
[Tk2] Tkachuk V.V.: A $C_p$-theory Problem Book. Topological and Function Spaces.Springer, New York, 2011. Zbl 1222.54002 |
| . |
