Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
filter convergence; ultrafilter; product; subproduct; sequential compactness; sequencewise $\mathcal P$-compactness; Lindelöf property; final $\lambda$-compactness; $[ \mu, \lambda ]$-compactness; Menger property; Rothberger property
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.
References:
[1] Blass A.: Combinatorial cardinal characteristics of the continuum. in Handbook of Set Theory, Springer, Dordrecht, 2010, pages 395–489. MR 2768685
[2] Booth D.: A Boolean view of sequential compactness. Fund. Math. 85 (1974), no. 2, 99–102. DOI 10.4064/fm-85-2-99-102 | MR 0367926
[3] Brandhorst S.: Tychonoff-Like Theorems and Hypercompact Topological Spaces. Bachelor's Thesis, Leibniz Universität, Hannover, 2013.
[4] Brandhorst S., Erné M.: Tychonoff-like product theorems for local topological properties. Topology Proc. 45 (2015), 121–138. MR 3231433
[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. MR 1739865 | Zbl 0955.03044
[6] Comfort W. W.: Article Review: Some applications of ultrafilters in topology. MathSciNet Mathematical Reviews 52 (1976), \# 1633, 227. MR 0451187
[7] van Douwen E. K.: The integers and topology. in Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 111–167. MR 0776622 | Zbl 0561.54004
[8] García-Ferreira S.: On FU($p$)-spaces and $p$-sequential spaces. Comment. Math. Univ. Carolin. 32 (1991), no. 1, 161–171. MR 1118299 | Zbl 0789.54032
[9] García-Ferreira S., Kočinac L.: Convergence with respect to ultrafilters: a survey. Filomat 10 (1996), 1–32. MR 1448484
[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. MR 1975381 | Zbl 1088.06001
[11] Ginsburg J., Saks V.: Some applications of ultrafilters in topology. Pacific J. Math. 57 (1975), no. 2, 403–418. DOI 10.2140/pjm.1975.57.403 | MR 0380736 | Zbl 0288.54020
[12] Goubault-Larrecq J.: Non-Hausdorff Topology and Domain Theory. New Mathematical Monographs, 22, Cambridge University Press, Cambridge, 2013. MR 3086734
[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
[14] Lipparini P.: Compact factors in finally compact products of topological spaces. Topology Appl. 153 (2006), no. 9, 1365–1382. DOI 10.1016/j.topol.2005.04.002 | MR 2211205 | Zbl 1093.54001
[15] Lipparini P.: A very general covering property. Comment. Math. Univ. Carolin. 53 (2012), no. 2, 281–306. MR 3017260
[16] Lipparini P.: A characterization of the Menger property by means of ultrafilter convergence. Topology Appl. 160 (2013), no. 18, 2505–2513. DOI 10.1016/j.topol.2013.07.044 | MR 3120664
[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
[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
[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
[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
[21] Nyikos P.: Sequential extensions of countably compact spaces. Topol. Proc. 31 (2007), no. 2, 651–665. MR 2476634
[22] Saks V.: Ultrafilter invariants in topological spaces. Trans. Amer. Math. Soc. 241 (1978), 79–97. DOI 10.1090/S0002-9947-1978-0492291-9 | MR 0492291
[23] Scarborough C. T., Stone A. H.: Products of nearly compact spaces. Trans. Amer. Math. Soc. 124 (1966), 131–147. DOI 10.1090/S0002-9947-1966-0203679-7 | MR 0203679
[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. MR 0776632 | Zbl 0588.54025
[25] Stephenson R. M., Jr., Vaughan J. E.: Products of initially $m$-compact spaces. Trans. Amer. Math. Soc. 196 (1974), 177–189. MR 0425898
[26] Usuba T.: $G_\delta$-topology and compact cardinals. Fund. Math. 246 (2019), no. 1, 71–87. DOI 10.4064/fm487-7-2018 | MR 3937917
[27] Vaughan J. E.: Countably compact and sequentially compact spaces. in Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 569–602. MR 0776631 | Zbl 0562.54031
[28] Vickers S.: Topology via Logic. Cambridge Tracts in Theoretical Computer Science, 5, Cambridge University Press, Cambridge, 1989. MR 1002193 | Zbl 0922.54002
Partner of
EuDML logo