| Title:
|
On butterfly-points in $\beta X$, Tychonoff products and weak Lindelöf numbers (English) |
| Author:
|
Logunov, Sergei |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
379-383 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Butterfly-point |
| Keyword:
|
non-normality point |
| Keyword:
|
Čech--Stone compactification |
| Keyword:
|
Tychonoff product |
| Keyword:
|
weak Lindelöf number |
| MSC:
|
54D15 |
| MSC:
|
54D35 |
| MSC:
|
54D40 |
| MSC:
|
54D80 |
| MSC:
|
54E35 |
| MSC:
|
54G20 |
| idZBL:
|
Zbl 07655807 |
| idMR:
|
MR4542796 |
| DOI:
|
10.14712/1213-7243.2022.023 |
| . |
| Date available:
|
2023-02-01T12:12:05Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151483 |
| . |
| Reference:
|
[1] Bešlagić A., van Douwen E. K.: Spaces of nonuniform ultrafilters in space of uniform ultrafilters.Topology Appl. 35 (1990), no. 2–3, 253–260. MR 1058805, 10.1016/0166-8641(90)90110-N |
| Reference:
|
[2] Błaszczyk A., Szymański A.: Some non-normal subspaces of the Čech–Stone compactification of a discrete space.Proc. Eighth Winter School on Abstract Analysis and Topology, Praha, 1980, Czechoslovak Academy of Sciences, Praha, 1980, pages 35–38. |
| Reference:
|
[3] Fine N. J., Gillman L.: Extension of continuous functions in $\beta N$.Bull. Amer. Math. Soc. 66 (1960), 376–381. MR 0123291, 10.1090/S0002-9904-1960-10460-0 |
| Reference:
|
[4] Logunov S.: On non-normality points and metrizable crowded spaces.Comment. Math. Univ. Carolin. 48 (2007), no. 3, 523–527. MR 2374131 |
| Reference:
|
[5] Logunov S.: Non-normality points and big products of metrizable spaces.Topology Proc. 46 (2015), 73–85. MR 3218260 |
| Reference:
|
[6] Logunov S.: On non-normality points, Tychonoff products and Suslin number.Comment. Math. Univ. Carolin. 63 (2022), no. 1, 131–134. MR 4445740 |
| Reference:
|
[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. MR 0394609 |
| Reference:
|
[8] Terasawa J.: $\beta X-\{p\}$ are non-normal for non-discrete spaces $X$.Topology Proc. 31 (2007), no. 1, 309–317. |
| Reference:
|
[9] Warren N. M.: Properties of Stone–Čech compactifications of discrete spaces.Proc. Amer. Math. Soc. 33 (1972), 599–606. |
| . |
