Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 3, page 397-408
  • ISSN: 0010-2628

Abstract

top
Given a Tychonoff space X and an infinite cardinal κ , we prove that exponential κ -domination in X is equivalent to exponential κ -cofinality of C p ( X ) . On the other hand, exponential κ -cofinality of X is equivalent to exponential κ -domination in C p ( X ) . We show that every exponentially κ -cofinal space X has a κ + -small diagonal; besides, if X is κ -stable, then n w ( X ) ≤ κ . In particular, any compact exponentially κ -cofinal space has weight not exceeding κ . We also establish that any exponentially κ -cofinal space X with l ( X ) ≤ κ and t ( X ) ≤ κ has i -weight not exceeding κ while for any cardinal κ , there exists an exponentially ø -cofinal space X such that l ( X ) ≥ κ .

How to cite

top

Tkachuk, Vladimir Vladimirovich. "Exponential domination in function spaces." Commentationes Mathematicae Universitatis Carolinae 61.3 (2020): 397-408. <http://eudml.org/doc/297383>.

@article{Tkachuk2020,
abstract = {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) \le \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) \le \kappa $ and $t(X) \le \kappa $ has $i$-weight not exceeding $\kappa $ while for any cardinal $\kappa $, there exists an exponentially $ø$-cofinal space $X$ such that $l(X) \ge \kappa $.},
author = {Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {exponential $\kappa $-domination; exponential $\kappa $-cofinality; $\kappa $-stable space; $i$-weight; function space; duality; $\kappa ^+$-small diagonal},
language = {eng},
number = {3},
pages = {397-408},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Exponential domination in function spaces},
url = {http://eudml.org/doc/297383},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Tkachuk, Vladimir Vladimirovich
TI - Exponential domination in function spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 3
SP - 397
EP - 408
AB - 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) \le \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) \le \kappa $ and $t(X) \le \kappa $ has $i$-weight not exceeding $\kappa $ while for any cardinal $\kappa $, there exists an exponentially $ø$-cofinal space $X$ such that $l(X) \ge \kappa $.
LA - eng
KW - exponential $\kappa $-domination; exponential $\kappa $-cofinality; $\kappa $-stable space; $i$-weight; function space; duality; $\kappa ^+$-small diagonal
UR - http://eudml.org/doc/297383
ER -

References

top
  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). MR0670475
  2. Arkhangel'skiĭ A. V., Continuous mappings, factorization theorems and spaces of functions, Trudy Moskov. Mat. Obshch. 47 (1984), 3–21, 246 (Russian). MR0774944
  3. Arkhangel'skiĭ A. V., 10.1007/978-94-011-2598-7_4, Mathematics and Its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. MR1144519DOI10.1007/978-94-011-2598-7_4
  4. Asanov M. O., On cardinal invariants of spaces of continuous functions, Sovr. Topologia i Teoria Mnozhestv 2 (1979), 8–12 (Russian). 
  5. Engelking R., General Topology,, Monografie Matematyczne, 60, PWN—Polish Scientific Publishers, Warsaw, 1977. MR0500780
  6. Gruenhage G., Tkachuk V. V., Wilson R. G., 10.1016/j.topol.2020.107306, Topology Appl. 282 (2020), 107306, 10 pages. MR4116835DOI10.1016/j.topol.2020.107306
  7. Hodel R. E., Cardinal Functions. I., Handbook of Set-Theoretic Topology, North Holland, Amsterdam, 1984, 1–61. MR0776620
  8. Hušek M., Topological spaces without κ -accessible diagonal, Comment. Math. Univ. Carolinae 18 (1977), no. 4, 777–788. MR0515009
  9. Juhász I., Szentmiklóssy Z., 10.2307/2159502, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1153–1160. Zbl0767.54002MR1137223DOI10.2307/2159502
  10. Noble N., 10.1090/S0002-9939-1974-0328855-4, Proc. Amer. Math. Soc. 42 (1974), no. 1, 228–233. MR0328855DOI10.1090/S0002-9939-1974-0328855-4
  11. Pytkeev E. G., Tightness of spaces of continuous functions, Uspekhi Mat. Nauk 37 (1982), no. 1(223), 157–158 (Russian). MR0643782
  12. Tkachuk V. V., A C p -Theory Problem Book, Topological and Function Spaces, Problem Books in Mathematics, Springer, New York, 2011. MR3024898
  13. Tkachuk V. V., A C p -Theory Problem Book, Special Features of Function Spaces, Problem Books in Mathematics, Springer, Cham, 2014. MR3243753
  14. Tkachuk V. V., A C p -Theory Problem Book, Compactness in Function Spaces, Problem Books in Mathematics, Springer, Cham, 2015. MR3243753

NotesEmbed ?

top

You must be logged in to post comments.