| Title:
|
A nice subclass of functionally countable spaces (English) |
| Author:
|
Tkachuk, Vladimir V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
399-409 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A space $X$ is {functionally countable} if $f(X)$ is countable for any continuous function $f\colon X \to {\mathbb{R}}$. 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 $A\cap \bigcap {\mathcal{G}}\neq\emptyset$ whenever ${\mathcal{G}}\subset {\mathcal{F}}$ and $\bigcap {\mathcal{G}}\neq\emptyset$. 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. (English) |
| Keyword:
|
countably compact space |
| Keyword:
|
Lindelöf space |
| Keyword:
|
Lindelöf $P$-space |
| Keyword:
|
functionally countable space |
| Keyword:
|
exponentially separable space |
| Keyword:
|
retraction |
| Keyword:
|
scattered space |
| Keyword:
|
extent |
| Keyword:
|
Sokolov space |
| Keyword:
|
weakly Sokolov space |
| Keyword:
|
function space |
| MSC:
|
54C35 |
| MSC:
|
54D65 |
| MSC:
|
54G10 |
| MSC:
|
54G12 |
| idZBL:
|
Zbl 06940880 |
| idMR:
|
MR3861562 |
| DOI:
|
10.14712/1213-7243.2015.254 |
| . |
| Date available:
|
2018-09-10T12:20:07Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147407 |
| . |
| Reference:
|
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| Reference:
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| . |
