On butterfly-points in β X , Tychonoff products and weak Lindelöf numbers

Sergei Logunov

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 3, page 379-383
  • ISSN: 0010-2628

Abstract

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Let X be the Tychonoff product ∏ α < τ X α of τ -many Tychonoff non-single point spaces X α . Let p ∈ X * be a point in the closure of some G ⊂ X whose weak Lindelöf number is strictly less than the cofinality of τ . Then we show that β X ∖ { p } is not normal. Under some additional assumptions, p is a butterfly-point in β X . In particular, this is true if either X = ω τ or X = R τ and τ is infinite and not countably cofinal.

How to cite

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Logunov, Sergei. "On butterfly-points in $\beta X$, Tychonoff products and weak Lindelöf numbers." Commentationes Mathematicae Universitatis Carolinae 62 63.3 (2022): 379-383. <http://eudml.org/doc/299036>.

@article{Logunov2022,
abstract = {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 \lbrace p\rbrace $ 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.},
author = {Logunov, Sergei},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Butterfly-point; non-normality point; Čech--Stone compactification; Tychonoff product; weak Lindelöf number},
language = {eng},
number = {3},
pages = {379-383},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On butterfly-points in $\beta X$, Tychonoff products and weak Lindelöf numbers},
url = {http://eudml.org/doc/299036},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Logunov, Sergei
TI - On butterfly-points in $\beta X$, Tychonoff products and weak Lindelöf numbers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 3
SP - 379
EP - 383
AB - 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 \lbrace p\rbrace $ 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.
LA - eng
KW - Butterfly-point; non-normality point; Čech--Stone compactification; Tychonoff product; weak Lindelöf number
UR - http://eudml.org/doc/299036
ER -

References

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  6. Logunov S., On non-normality points, Tychonoff products and Suslin number, Comment. Math. Univ. Carolin. 63 (2022), no. 1, 131–134. MR4445740
  7. Shapirovskij B., On embedding extremely disconnected spaces in compact Hausdorff spaces, b -points and weight of pointwise normal spaces, Dokl. Akad. Nauk SSSR 223 (1987), 1083–1086. MR0394609
  8. Terasawa J., β X - { p } are non-normal for non-discrete spaces X , Topology Proc. 31 (2007), no. 1, 309–317. 
  9. Warren N. M., Properties of Stone–Čech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599–606. 

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