On butterfly-points in $\beta X$, Tychonoff products and weak Lindelöf numbers

Logunov, Sergei

Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 3, pp. 379-383.

Résumé

Let $X$ be the Tychonoff product $\prod _{\alpha \tau}X_{\alpha}$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha}$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \{p\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X=\omega^{\tau}$ or $X=R^{\tau}$ and $\tau$ is infinite and not countably cofinal.

Détail

DOI : 10.14712/1213-7243.2022.023

Classification : 54D15, 54D35, 54D40, 54D80, 54E35, 54G20
Keywords: Butterfly-point; non-normality point; Čech--Stone compactification; Tychonoff product; weak Lindelöf number