Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (2020)

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

Abstract

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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

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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

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