On non-normality points, Tychonoff products and Suslin number

Sergei Logunov

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 1, page 131-134
  • ISSN: 0010-2628

Abstract

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Let a space X be Tychonoff product ∏ α < τ X α of τ -many Tychonoff nonsingle point spaces X α . Let Suslin number of X be strictly less than the cofinality of τ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification β X . In particular, this is true if X is either R τ or ω τ and a cardinal τ is infinite and not countably cofinal.

How to cite

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Logunov, Sergei. "On non-normality points, Tychonoff products and Suslin number." Commentationes Mathematicae Universitatis Carolinae 62 63.1 (2022): 131-134. <http://eudml.org/doc/299271>.

@article{Logunov2022,
abstract = {Let a space $X$ be Tychonoff product $\prod _\{\alpha <\tau \}X_\{\alpha \}$ of $\tau $-many Tychonoff nonsingle point spaces $X_\{\alpha \}$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau $. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^\{\tau \}$ or $\omega ^\{\tau \}$ and a cardinal $\tau $ is infinite and not countably cofinal.},
author = {Logunov, Sergei},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {non-normality point; Čech--Stone compactification; Tychonoff product; Suslin number},
language = {eng},
number = {1},
pages = {131-134},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On non-normality points, Tychonoff products and Suslin number},
url = {http://eudml.org/doc/299271},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Logunov, Sergei
TI - On non-normality points, Tychonoff products and Suslin number
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 1
SP - 131
EP - 134
AB - Let a space $X$ be Tychonoff product $\prod _{\alpha <\tau }X_{\alpha }$ of $\tau $-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau $. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or $\omega ^{\tau }$ and a cardinal $\tau $ is infinite and not countably cofinal.
LA - eng
KW - non-normality point; Čech--Stone compactification; Tychonoff product; Suslin number
UR - http://eudml.org/doc/299271
ER -

References

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  1. Blaszczyk A., Szymański A., Some non-normal subspaces of the Čech–Stone compactification of a discrete space, Abstracta, Eighth Winter School on Abstract Analysis, Czechoslovak Academy of Sciences, Praha, 1980, pages 35–38. 
  2. Bešlagić A., van Douwen E. K., 10.1016/0166-8641(90)90110-N, Topology Appl. 35 (1990), no. 2–3, 253–260. MR1058805DOI10.1016/0166-8641(90)90110-N
  3. Fine N. J., Gillman L., 10.1090/S0002-9904-1960-10460-0, Bull. Amer. Math. Soc. 66 (1960), 376–381. MR0123291DOI10.1090/S0002-9904-1960-10460-0
  4. Logunov S., On non-normality points and metrizable crowded spaces, Comment. Math. Univ. Carolin. 48 (2007), no. 3, 523–527. MR2374131
  5. Logunov S., Non-normality points and big products of metrizable spaces, Topology Proc. 46 (2015), 73–85. MR3218260
  6. Terasawa J., β X - { p } are non-normal for non-discrete spaces X , Topology Proc. 31 (2007), no. 1, 309–317. MR2363172
  7. Warren N. M., Properties of Stone–Čech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599–606. MR0292035

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