A nice subclass of functionally countable spaces

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 3, page 399-409
  • ISSN: 0010-2628

Abstract

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A space X is functionally countable if f ( X ) is countable for any continuous function f : X → ℝ . We will call a space X exponentially separable if for any countable family ℱ of closed subsets of X , there exists a countable set A ⊂ X such that A ∩ ⋂ 𝒢 ≠ ∅ whenever 𝒢 ⊂ ℱ and ⋂ 𝒢 ≠ ∅ . Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has nice categorical properties: it is preserved by closed subspaces, countable unions and continuous images. Besides, it contains all Lindelöf P -spaces as well as some wide classes of scattered spaces. In particular, if a scattered space is either Lindelöf or ω -bounded, then it is exponentially separable.

How to cite

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Tkachuk, Vladimir Vladimirovich. "A nice subclass of functionally countable spaces." Commentationes Mathematicae Universitatis Carolinae 59.3 (2018): 399-409. <http://eudml.org/doc/294597>.

@article{Tkachuk2018,
abstract = {A space $X$ is functionally countable if $f(X)$ is countable for any continuous function $f\colon X \rightarrow \{\mathbb \{R\}\}$. We will call a space $X$ exponentially separable if for any countable family $\{\mathcal \{F\}\}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap \{\mathcal \{G\}\}\ne \emptyset $ whenever $\{\mathcal \{G\}\}\subset \{\mathcal \{F\}\}$ and $\bigcap \{\mathcal \{G\}\}\ne \emptyset $. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has nice categorical properties: it is preserved by closed subspaces, countable unions and continuous images. Besides, it contains all Lindelöf $P$-spaces as well as some wide classes of scattered spaces. In particular, if a scattered space is either Lindelöf or $\{\omega \}$-bounded, then it is exponentially separable.},
author = {Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {countably compact space; Lindelöf space; Lindelöf $P$-space; functionally countable space; exponentially separable space; retraction; scattered space; extent; Sokolov space; weakly Sokolov space; function space},
language = {eng},
number = {3},
pages = {399-409},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A nice subclass of functionally countable spaces},
url = {http://eudml.org/doc/294597},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Tkachuk, Vladimir Vladimirovich
TI - A nice subclass of functionally countable spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 3
SP - 399
EP - 409
AB - A space $X$ is functionally countable if $f(X)$ is countable for any continuous function $f\colon X \rightarrow {\mathbb {R}}$. We will call a space $X$ exponentially separable if for any countable family ${\mathcal {F}}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap {\mathcal {G}}\ne \emptyset $ whenever ${\mathcal {G}}\subset {\mathcal {F}}$ and $\bigcap {\mathcal {G}}\ne \emptyset $. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has nice categorical properties: it is preserved by closed subspaces, countable unions and continuous images. Besides, it contains all Lindelöf $P$-spaces as well as some wide classes of scattered spaces. In particular, if a scattered space is either Lindelöf or ${\omega }$-bounded, then it is exponentially separable.
LA - eng
KW - countably compact space; Lindelöf space; Lindelöf $P$-space; functionally countable space; exponentially separable space; retraction; scattered space; extent; Sokolov space; weakly Sokolov space; function space
UR - http://eudml.org/doc/294597
ER -

References

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