| Title:
|
Exponential domination in function spaces (English) |
| Author:
|
Tkachuk, Vladimir V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
397-408 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $\,C_p(X)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p(X)$. We show that every exponentially $\kappa$-cofinal space $X$ has a $\kappa^+$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw(X) \leq \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal space $X$ with $l(X) \leq\kappa$ and $t(X) \leq \kappa$ has $i$-weight not exceeding $\kappa$ while for any cardinal $\kappa$, there exists an exponentially $\o$-cofinal space $X$ such that $l(X) \geq \kappa$. (English) |
| Keyword:
|
exponential $\kappa$-domination |
| Keyword:
|
exponential $\kappa$-cofinality |
| Keyword:
|
$\kappa$-stable space |
| Keyword:
|
$i$-weight |
| Keyword:
|
function space |
| Keyword:
|
duality |
| Keyword:
|
$\kappa^+$-small diagonal |
| MSC:
|
54C05 |
| MSC:
|
54C35 |
| MSC:
|
54G20 |
| idZBL:
|
Zbl 07286012 |
| idMR:
|
MR4186115 |
| DOI:
|
10.14712/1213-7243.2020.032 |
| . |
| Date available:
|
2020-11-27T07:47:01Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148474 |
| . |
| Reference:
|
[1] Arkhangel'skiĭ A. V.: Factorization theorems and spaces of functions: stability and monolithism.Dokl. Akad. Nauk SSSR 265 (1982), no. 5, 1039–1043 (Russian). MR 0670475 |
| Reference:
|
[2] Arkhangel'skiĭ A. V.: Continuous mappings, factorization theorems and spaces of functions.Trudy Moskov. Mat. Obshch. 47 (1984), 3–21, 246 (Russian). MR 0774944 |
| Reference:
|
[3] Arkhangel'skiĭ A. V.: Topological Function Spaces.Mathematics and Its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1144519, 10.1007/978-94-011-2598-7_4 |
| Reference:
|
[4] Asanov M. O.: On cardinal invariants of spaces of continuous functions.Sovr. Topologia i Teoria Mnozhestv 2 (1979), 8–12 (Russian). |
| Reference:
|
[5] Engelking R.: General Topology,.Monografie Matematyczne, 60, PWN—Polish Scientific Publishers, Warsaw, 1977. MR 0500780 |
| Reference:
|
[6] Gruenhage G., Tkachuk V. V., Wilson R. G.: Domination by small sets versus density.Topology Appl. 282 (2020), 107306, 10 pages. MR 4116835, 10.1016/j.topol.2020.107306 |
| Reference:
|
[7] Hodel R. E.: Cardinal Functions. I..Handbook of Set-Theoretic Topology, North Holland, Amsterdam, 1984, 1–61. MR 0776620 |
| Reference:
|
[8] Hušek M.: Topological spaces without $\kappa$-accessible diagonal.Comment. Math. Univ. Carolinae 18 (1977), no. 4, 777–788. MR 0515009 |
| Reference:
|
[9] Juhász I., Szentmiklóssy Z.: Convergent free sequences in compact spaces.Proc. Amer. Math. Soc. 116 (1992), no. 4, 1153–1160. Zbl 0767.54002, MR 1137223, 10.2307/2159502 |
| Reference:
|
[10] Noble N.: The density character of function spaces.Proc. Amer. Math. Soc. 42 (1974), no. 1, 228–233. MR 0328855, 10.1090/S0002-9939-1974-0328855-4 |
| Reference:
|
[11] Pytkeev E. G.: Tightness of spaces of continuous functions.Uspekhi Mat. Nauk 37 (1982), no. 1(223), 157–158 (Russian). MR 0643782 |
| Reference:
|
[12] Tkachuk V. V.: A $C_p$-Theory Problem Book.Topological and Function Spaces, Problem Books in Mathematics, Springer, New York, 2011. MR 3024898 |
| Reference:
|
[13] Tkachuk V. V.: A $C_p$-Theory Problem Book.Special Features of Function Spaces, Problem Books in Mathematics, Springer, Cham, 2014. MR 3243753 |
| Reference:
|
[14] Tkachuk V. V.: A $C_p$-Theory Problem Book.Compactness in Function Spaces, Problem Books in Mathematics, Springer, Cham, 2015. MR 3243753 |
| . |
