Article
| Title: | Products of topological spaces and families of filters (English) |
| Author: | Lipparini, Paolo |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 64 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 373-394 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. We prove that a product is Lindelöf if and only if all subproducts by $\leq \omega_1 $ factors are Lindelöf. Parallel results are obtained for final $ \omega_n$-compactness, $[ \lambda, \mu ]$-compactness, the Menger and the Rothberger properties. (English) |
| Keyword: | filter convergence |
| Keyword: | ultrafilter |
| Keyword: | product |
| Keyword: | subproduct |
| Keyword: | sequential compactness |
| Keyword: | sequencewise $\mathcal P$-compactness |
| Keyword: | Lindelöf property |
| Keyword: | final $\lambda$-compactness |
| Keyword: | $[ \mu, \lambda ]$-compactness |
| Keyword: | Menger property |
| Keyword: | Rothberger property |
| MSC: | 54A20 |
| MSC: | 54B10 |
| MSC: | 54D20 |
| idZBL: | Zbl 07830515 |
| idMR: | MR4717508 |
| DOI: | 10.14712/1213-7243.2024.005 |
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| Date available: | 2024-03-18T10:47:28Z |
| Last updated: | 2024-08-02 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152305 |
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| Reference: | [1] Blass A.: Combinatorial cardinal characteristics of the continuum.in Handbook of Set Theory, Springer, Dordrecht, 2010, pages 395–489. MR 2768685 |
| Reference: | [2] Booth D.: A Boolean view of sequential compactness.Fund. Math. 85 (1974), no. 2, 99–102. MR 0367926, 10.4064/fm-85-2-99-102 |
| Reference: | [3] Brandhorst S.: Tychonoff-Like Theorems and Hypercompact Topological Spaces.Bachelor's Thesis, Leibniz Universität, Hannover, 2013. |
| Reference: | [4] Brandhorst S., Erné M.: Tychonoff-like product theorems for local topological properties.Topology Proc. 45 (2015), 121–138. MR 3231433 |
| Reference: | [5] Caicedo X.: The abstract compactness theorem revisited.in Logic and Foundations of Mathematics, Synthese Lib., 280, Kluwer Acad. Publ., Dordrecht, 1999, pages 131–141. Zbl 0955.03044, MR 1739865 |
| Reference: | [6] Comfort W. W.: Article Review: Some applications of ultrafilters in topology.MathSciNet Mathematical Reviews 52 (1976), \# 1633, 227. MR 0451187 |
| Reference: | [7] van Douwen E. K.: The integers and topology.in Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 111–167. Zbl 0561.54004, MR 0776622 |
| Reference: | [8] García-Ferreira S.: On FU($p$)-spaces and $p$-sequential spaces.Comment. Math. Univ. Carolin. 32 (1991), no. 1, 161–171. Zbl 0789.54032, MR 1118299 |
| Reference: | [9] García-Ferreira S., Kočinac L.: Convergence with respect to ultrafilters: a survey.Filomat 10 (1996), 1–32. MR 1448484 |
| Reference: | [10] Gierz G., Hofmann K. H., Keimel K., Lawson J. D., Mislove M., Scott D. S.: Continuous Lattices and Domains.Encyclopedia of Mathematics and Its Applications, 93, Cambridge University Press, Cambridge, 2003. Zbl 1088.06001, MR 1975381 |
| Reference: | [11] Ginsburg J., Saks V.: Some applications of ultrafilters in topology.Pacific J. Math. 57 (1975), no. 2, 403–418. Zbl 0288.54020, MR 0380736, 10.2140/pjm.1975.57.403 |
| Reference: | [12] Goubault-Larrecq J.: Non-Hausdorff Topology and Domain Theory.New Mathematical Monographs, 22, Cambridge University Press, Cambridge, 2013. MR 3086734 |
| Reference: | [13] Kombarov A. P.: Compactness and sequentiality with respect to a set of ultrafilters.Vestnik Moskov. Univ. Ser. I Mat. Mekh. 95 (1985), no. 5, 15–18 (Russian); translation in Moscow Univ. Math. Bull. 40 (1985), no. 5, 15–18. MR 0814266 |
| Reference: | [14] Lipparini P.: Compact factors in finally compact products of topological spaces.Topology Appl. 153 (2006), no. 9, 1365–1382. Zbl 1093.54001, MR 2211205, 10.1016/j.topol.2005.04.002 |
| Reference: | [15] Lipparini P.: A very general covering property.Comment. Math. Univ. Carolin. 53 (2012), no. 2, 281–306. MR 3017260 |
| Reference: | [16] Lipparini P.: A characterization of the Menger property by means of ultrafilter convergence.Topology Appl. 160 (2013), no. 18, 2505–2513. MR 3120664, 10.1016/j.topol.2013.07.044 |
| Reference: | [17] Lipparini P.: Topological spaces compact with respect to a set of filters.Cent. Eur. J. Math. 12 (2014), no. 7, 991–999. MR 3188459 |
| Reference: | [18] Lipparini P.: Products of sequentially compact spaces with no separability assumption.Rend. Istit. Mat. Univ. Trieste 54 (2022), Art. No. 8, 9 pages. MR 4595165 |
| Reference: | [19] Lipparini P.: Products of sequentially compact spaces and compactness with respect to a set of filters.available at arXiv:1303.0815v5 [math.GN] (2022), 32 pages. MR 3188459 |
| Reference: | [20] Mycielski I.: Two remarks on Tychonoff's product theorem.Bull. Acad. Polon. Sci. Sér. Sci. Math., Astronom. Phys. 12 (1964), 439–441. MR 0215731 |
| Reference: | [21] Nyikos P.: Sequential extensions of countably compact spaces.Topol. Proc. 31 (2007), no. 2, 651–665. MR 2476634 |
| Reference: | [22] Saks V.: Ultrafilter invariants in topological spaces.Trans. Amer. Math. Soc. 241 (1978), 79–97. MR 0492291, 10.1090/S0002-9947-1978-0492291-9 |
| Reference: | [23] Scarborough C. T., Stone A. H.: Products of nearly compact spaces.Trans. Amer. Math. Soc. 124 (1966), 131–147. MR 0203679, 10.1090/S0002-9947-1966-0203679-7 |
| Reference: | [24] Stephenson R. M., Jr.: Initially $\kappa$-compact and related spaces.in Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 603–632. Zbl 0588.54025, MR 0776632 |
| Reference: | [25] Stephenson R. M., Jr., Vaughan J. E.: Products of initially $m$-compact spaces.Trans. Amer. Math. Soc. 196 (1974), 177–189. MR 0425898 |
| Reference: | [26] Usuba T.: $G_\delta$-topology and compact cardinals.Fund. Math. 246 (2019), no. 1, 71–87. MR 3937917, 10.4064/fm487-7-2018 |
| Reference: | [27] Vaughan J. E.: Countably compact and sequentially compact spaces.in Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 569–602. Zbl 0562.54031, MR 0776631 |
| Reference: | [28] Vickers S.: Topology via Logic.Cambridge Tracts in Theoretical Computer Science, 5, Cambridge University Press, Cambridge, 1989. Zbl 0922.54002, MR 1002193 |
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