Exponential separability is preserved by some products

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 3, page 385-395
  • ISSN: 0010-2628

Abstract

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We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a σ -compact crowded space in which all countable subspaces are scattered. If X is a Lindelöf space and every Y ⊂ X with | Y | ≤ 2 ω 1 is scattered, then X is functionally countable; if every Y ⊂ X with | Y | ≤ 2 𝔠 is scattered, then X is exponentially separable. A Lindelöf Σ -space X must be exponentially separable provided that every Y ⊂ X with | Y | ≤ 𝔠 is scattered. Under the Luzin axiom ( 2 ω 1 > 𝔠 ) we characterize weak exponential separability of C p ( X , [ 0 , 1 ] ) for any metrizable space X . Our results solve several published open questions.

How to cite

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Tkachuk, Vladimir Vladimirovich. "Exponential separability is preserved by some products." Commentationes Mathematicae Universitatis Carolinae 62 63.3 (2022): 385-395. <http://eudml.org/doc/298966>.

@article{Tkachuk2022,
abstract = {We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma $-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\le 2^\{\omega _1\}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\le 2^\{\mathfrak \{c\}\} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma $-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\le \{\mathfrak \{c\}\}$ is scattered. Under the Luzin axiom ($2^\{\omega _1\}>\{\mathfrak \{c\}\} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions.},
author = {Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lindelöf space; scattered space; $\sigma $-product; function space; $P$-space; exponentially separable space; product; functionally countable space; weakly exponentially separable space},
language = {eng},
number = {3},
pages = {385-395},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Exponential separability is preserved by some products},
url = {http://eudml.org/doc/298966},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Tkachuk, Vladimir Vladimirovich
TI - Exponential separability is preserved by some products
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 3
SP - 385
EP - 395
AB - We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma $-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\le 2^{\omega _1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\le 2^{\mathfrak {c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma $-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\le {\mathfrak {c}}$ is scattered. Under the Luzin axiom ($2^{\omega _1}>{\mathfrak {c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions.
LA - eng
KW - Lindelöf space; scattered space; $\sigma $-product; function space; $P$-space; exponentially separable space; product; functionally countable space; weakly exponentially separable space
UR - http://eudml.org/doc/298966
ER -

References

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