Some remarks on Pixley-Roy hyperspaces
In this paper, we study cellular-compact, cellular-Lindel\"of, strongly star-Hurewicz, strongly star-Rothberger, strongly star-Menger spaces on hyperspaces with the Pixley-Roy topology. For a space $X$ and $n\in\mathbb N$, we prove that (1) If $\texttt{PR}[X]$ or $\texttt{PR}_n[X]$ is cellular...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
HUJOS
2025-06-01
|
| Series: | Tạp chí Khoa học Đại học Huế: Khoa học Tự nhiên |
| Subjects: | |
| Online Access: | https://jos.hueuni.edu.vn/index.php/hujos-ns/article/view/7340 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | In this paper, we study cellular-compact, cellular-Lindel\"of, strongly star-Hurewicz, strongly star-Rothberger, strongly star-Menger spaces on hyperspaces with the Pixley-Roy topology. For a space $X$ and $n\in\mathbb N$, we prove that
(1) If $\texttt{PR}[X]$ or $\texttt{PR}_n[X]$ is cellular-compact, then $X$ is cellular-compact. However, there exists a compact space $X$ such that $|X|=\omega$, but $\texttt{PR}_n[X]$ for all $n\in\mathbb N$ and $\texttt{PR}[X]$ are not cellular-compact spaces.
(2) If $\texttt{PR}[X]$ or $\texttt{PR}_n[X]$ is cellular-Lindel\"of, then $X$ is cellular-Lindel\"of.
(3) $X$ is countable if and only if $\texttt{PR}[X]$ is a strongly star-Hurewicz space, if and only if $\texttt{PR}[X]$ is a strongly star-Rothberger space if and only if$\texttt{PR}[X]$ is a strongly star-Menger space.
|
|---|---|
| ISSN: | 1859-1388 2615-9678 |