Characterizing function spaces which have the property (B) of Banakh
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929438181031936 |
|---|---|
| author | Krupski, Mikołaj Kucharski, Kacper Marciszewski, Witold |
| author_facet | Krupski, Mikołaj Kucharski, Kacper Marciszewski, Witold |
| contents | A topological space $Y$ has the property (B) of Banakh if there is a countable family $\{A_n:n\in \mathbb{N}\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space $X$ has the following property $(κ)$: every sequence of disjoint finite subsets of $X$ has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces $X$ whose all bounded subsets are finite, yet $X$ fails to have the property $(κ)$. This answers a question of Tkachuk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18618 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizing function spaces which have the property (B) of Banakh Krupski, Mikołaj Kucharski, Kacper Marciszewski, Witold General Topology 54C35, 54E52, 54A10 A topological space $Y$ has the property (B) of Banakh if there is a countable family $\{A_n:n\in \mathbb{N}\}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space $X$ has the following property $(κ)$: every sequence of disjoint finite subsets of $X$ has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces $X$ whose all bounded subsets are finite, yet $X$ fails to have the property $(κ)$. This answers a question of Tkachuk. |
| title | Characterizing function spaces which have the property (B) of Banakh |
| topic | General Topology 54C35, 54E52, 54A10 |
| url | https://arxiv.org/abs/2407.18618 |