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Keywords:
Kunen point; $\beta$-normality; Čech--Stone compactification; metrizable crowded space
Kunen point; $\beta$-normality; Čech--Stone compactification; metrizable crowded space
Summary:
We show that every point of the remainder $\beta X\setminus X$ of the Čech--Stone compactification $\beta X$ of any metrizable crowded space $X$ is a ``$\lambda$-Kunen" point for some regular cardinal $\lambda$. As a consequence we show that $\beta X \setminus \{p\}$ is not $\beta$-normal in the sense of result published in the paper On $\alpha$-normal and $\beta$-normal spaces (2021) by A. V. Arhangel'skii and L. Ludwig and, it explicitly indicate closed subsets of $\beta X \setminus \{p\}$ that cannot be ``$\beta$-separated".
References:
[1] Arhangel'skii A. V., Ludwig L.: On $\alpha$-normal and $\beta$-normal spaces. Comment. Math. Univ. Carolin. 42 (2001), no. 3, 507–519.
[2] Logunov S. A.: On non-normality points and metrizable crowded spaces. Comment. Math. Univ. Carolin. 48 (2007), no. 3, 523–527.
[3] Logunov S. A.: Non-normality points and nice spaces. Comment. Math. Univ. Carolin. 62 (2021), no. 3, 383–392.
[4] Terasawa J.: $\beta X-\{p\}$ are non-normal for non-discrete spaces $X$. Topology Proc. 31 (2007), no. 1, 309–317.
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