Article
| Title: | On Szymański theorem on hereditary normality of $\beta\omega$ (English) |
| Author: | Logunov, Sergei |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 507-512 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We discuss the following result of A. Szymański in ``Retracts and non-normality points" (2012), Corollary 3.5.: If $F$ is a closed subspace of $\omega ^{*}$ and the $\pi$-weight of $F$ is countable, then every nonisolated point of $F$ is a non-normality point of $\omega ^{*}$. We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in ``Some non-normal subspaces of the Čech--Stone compactification of a discrete space'' (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs look ``more natural in this area''. (English) |
| Keyword: | Čech--Stone compactification |
| Keyword: | non-normality point |
| Keyword: | butterfly-point |
| Keyword: | countable $\pi$-weight |
| MSC: | 54D15 |
| MSC: | 54D35 |
| MSC: | 54D40 |
| MSC: | 54D80 |
| MSC: | 54E35 |
| MSC: | 54G20 |
| idZBL: | Zbl 07729556 |
| idMR: | MR4577044 |
| DOI: | 10.14712/1213-7243.2023.011 |
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| Date available: | 2023-04-20T13:56:06Z |
| Last updated: | 2023-10-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151649 |
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| Reference: | [1] Bešlagić A., van Douwen E. K.: Spaces of nonuniform ultrafilters in spaces 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.Abstracta, 8th Winter School on Abstract Analysis, Praha, Czechoslovak Academy of Sciences, 1980, 35–38. |
| Reference: | [3] Gryzlov A. A.: On the question of hereditary normality of the space $\beta \omega \setminus \omega$.Topology and Set Theory Udmurt. Gos. Univ. Izhevsk (1982), 61–64 (Russian). MR 0760274 |
| 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] Rajagopalan M.: $\beta N-N-\{p\}$ is not normal.J. Indian Math. Soc. (N.S.) 36 (1972), 173–176. MR 0321012 |
| Reference: | [6] Shapirovkij B.: On embedding extremely disconnected spaces in compact Hausdorff spaces, $b$-points and weight of point-wise normal spaces.Dokl. Akad. Nauk SSSR 223 (1975), 1083–1086 (Russian). MR 0394609 |
| Reference: | [7] Szymański A.: Retracts and non-normality points.Topology Proc. 40 (2012), 195–201. MR 2832067 |
| Reference: | [8] Warren N. M.: Properties of Stone–Čech compactifications of discrete spaces.Proc. Amer. Math. Soc. 33 (1972), 599–606. MR 0292035 |
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