A remark on Pixley-Roy hyperspaces

Luong Quoc Tuyen ; Ong Van Tuyen 

Novi Sad Journal of Mathematics, Tome 54 (2024) no. 1.

Résumé

In this paper, we study the relation between a space $X$ satisfying certain generalized metric properties and the Pixley-Roy hyperspace $\texttt{PR}[X]$ over $X$ satisfying the same properties. We prove that a regular space $X$ is a countable stric $\mathfrak B_0$-space if and only if $\texttt{PR}_2[X]$ is a stric $\mathfrak B_0$-space. However, there exists a countable stric $\mathfrak B_0$-space $X$ such that $\texttt{PR}_n[X]$ with $n\geq 3$ and $\texttt{PR}[X]$ are not stric $\mathfrak B_0$-spaces. Moreover, we show that $\texttt{PR}[X]$ is a compact space if and only if $X$ is finite, and there exists a compact subset $K$ of a space $X$ such that $[\{x\},K]$ with $x\in K$ is not a compact subset of $\texttt{PR}[X]$. On the other hand, $X$ is a $P$-space if and only if so is $\texttt{PR}[X]$. Finally, we prove that if $\texttt{PR}[X]$ of a regular space $X$ is an $r$-space, then $X$ is also an $r$-space.

Détail

Publié le : 2022-07-18

DOI : 10.30755/NSJOM.14388

Classification : 54B20, 54D20, 54G10
Keywords: Pixley-Roy, hyperspace, stric $\mathfrak B_0$-space, compact space, $r$-space, $P$-space