Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 22, 2012

Sequential + separable vs sequentially separable and another variation on selective separability

  • Angelo Bella EMAIL logo , Maddalena Bonanzinga and Mikhail Matveev
From the journal Open Mathematics


A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

[1] Arhangel’skił A.V., Franklin S.P., Ordinal invariants for topological spaces, Michigan Math. J., 1968, 15, 313–320 in Google Scholar

[2] Barman D., Dow A., Selective separability and SS+, Topology Proc., 2011, 37, 181–204 Search in Google Scholar

[3] Bella A., More on sequential properties of 2ω1, Questions Answers Gen. Topology, 2004, 22(1), 1–4 Search in Google Scholar

[4] Bella A., Bonanzinga M., Matveev M., Variations of selective separability, Topology Appl., 2009, 156(7), 1241–1252 in Google Scholar

[5] Bella A., Bonanzinga M., Matveev M., Addendum to “Variations of selective separability” [Topology Appl., 156 (7) 2009, 1241–1252], Topology Appl., 2010, 157(15), 2389–2391 in Google Scholar

[6] Bella A., Bonanzinga M., Matveev M.V., Tkachuk V.V., Selective separability: general facts and behavior in countable spaces, In: Spring Topology and Dynamics Conference, Topology Proc., 2008, 32(Spring), 15–30 Search in Google Scholar

[7] Bella A., Matveev M., Spadaro S., Variations of selective separability II: Discrete sets and the influence of convergence and maximality, Topology Appl., 2012, 159(1), 253–271 in Google Scholar

[8] van Douwen E.K., The integers and topology, In: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 111–167 10.1016/B978-0-444-86580-9.50006-9Search in Google Scholar

[9] van Douwen E.K., Applications of maximal topologies, Topology Appl., 1993, 51(2), 125–139 in Google Scholar

[10] Dow A., Sequential order under MA, Topology Appl., 2005, 146/147, 501–510 in Google Scholar

[11] Dow A., Vaughan J.E., Ordinal remainders of classical -spaces, Fund. Math., 2012, 217(1), 83–93 in Google Scholar

[12] Engelking R., General Topology, Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989 Search in Google Scholar

[13] Gartside P., Lo J.T.H., Marsh A., Sequential density, Topology Appl., 2003, 130(1), 75–86 in Google Scholar

[14] Gruenhage G., Sakai M., Selective separability and its variations, Topology Appl., 2011, 158(12), 1352–1359 in Google Scholar

[15] Hrušák M., Steprāns J., Cardinal invariants related to sequential separability, In: Axiomatic Set Theory, Kyoto, November 15–17, 2000, Sūrikaisekikenkyūsho Kōkyūroku, 1202, Research Institute for Mathematical Sciences, Kyoto, 2001, 66–74 Search in Google Scholar

[16] Matveev M., Cardinal p and a theorem of Pelczynski, preprint available at Search in Google Scholar

[17] Miller A.W., Fremlin D.H., On some properties of Hurewicz, Menger, and Rothberger, Fund. Math., 1988, 129(1), 17–33 10.4064/fm-129-1-17-33Search in Google Scholar

[18] Scheepers M., Combinatorics of open covers I: Ramsey theory, Topology Appl., 1996, 69(1), 31–62 in Google Scholar

[19] Scheepers M., Combinatorics of open covers VI: Selectors for sequences of dense sets, Quaest. Math., 1999, 22(1), 109–130 in Google Scholar

[20] Tironi G., Isler R., On some problems of local approximability in compact spaces, In: General Topology and its Relations to Modern Analysis and Algebra, III, Prague, August 30–September 3, 1971, Academia, Prague, 1972, 443–446 Search in Google Scholar

[21] Vaughan J.E., Small uncountable cardinals and topology, In: Open Problems in Topology, North-Holland, Amsterdam, 1990, 195–218 Search in Google Scholar

[22] Velichko N.V., On sequential separability, Math. Notes, 2005, 78(5–6), 610–614 in Google Scholar

[23] Wilansky A., How separable is a space?, Amer. Math. Monthly, 1972, 79(7), 764–765 in Google Scholar

Published Online: 2012-12-22
Published in Print: 2013-3-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.3.2023 from
Scroll Up Arrow