×

Variations of selective separability II: Discrete sets and the influence of convergence and maximality. (English) Zbl 1239.54014

A topological space \(X\) is said to be selectively (\(R\)-) separable in case, for every sequence \(\{D_n: n\in\omega\}\) of dense subspaces \(D_n\), there exist finite (one point) subsets \(F_n\subset D_n\) such that \(\bigcup\{F_n: n\in\omega\}\) is dense in \(X\). Some variants of these notions are studied in the present paper. They have attracted recent attention as shown by the survey of G. Gruenhage and M. Sakai [“Selective separability and its variations”, Topology Appl. 158, No. 12, 1352–1359 (2011; Zbl 1228.54028)]. It was observed there that separable Fréchet spaces are \(R\)-separable. The present authors strengthen this result in various ways. A weaker version of selective separability is \(D\)-separability where \(F_n\) is merely assumed to be discrete. The authors show, among other things, that \(2^c\) is not \(D\)-separable. Various counterexamples are given, and a number of open questions and problems are posed.

MSC:

54D65 Separability of topological spaces
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D55 Sequential spaces
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)

Citations:

Zbl 1228.54028
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amirdzhanov, G., On dense subspaces of countable pseudocharacter and other generalizations of separability, Dokl. Akad. Nauk USSR, 234, 5, 993-996 (1977) · Zbl 0419.54002
[2] Arhangelʼskii, A. V., Bicompacta and the unions of countable families of metrizable subspaces, Dokl. Akad. Nauk USSR, 232, 5, 989-992 (1977)
[3] Arhangelʼskii, A. V., On d-separable spaces, (Seminar on General Topology (1981), Moscow State Univ. Publ.: Moscow State Univ. Publ. Moscow), 3-8 · Zbl 0858.54003
[4] Arhangelʼskii, A. V., Hurewicz spaces, analytic sets and fan-tightness of spaces of functions, Soviet Math. Dokl., 33, 2, 396-399 (1986) · Zbl 0606.54013
[5] Arhangelʼskii, A. V.; Bella, A., Countable fan tightness versus countable tightness, Comment. Math. Univ. Carolin., 37, 565-576 (1996) · Zbl 0881.54005
[6] Arhangelʼskii, A. V.; Collins, P. J., On submaximal spaces, Topology Appl., 64, 219-241 (1995) · Zbl 0826.54002
[7] Arhangelʼskii, A. V.; Franklin, S. P., Ordinal invariants for topological spaces, Michigan Math. J., 15, 313-320 (1968) · Zbl 0167.51102
[8] Aurichi, L., D-spaces, topological games, and selection principles, Topology Proc., 36, 107-122 (2010) · Zbl 1203.54024
[9] Babinkostova, L., On some questions about selective separability, Math. Log. Quart., 55, 539-541 (2009) · Zbl 1207.54035
[10] Barman, D.; Dow, A., Selective separability and \(SS^+\), Topology Proc., 37, 181-204 (2011) · Zbl 1207.54036
[11] D. Barman, A. Dow, More results on selective separability, preprint.; D. Barman, A. Dow, More results on selective separability, preprint. · Zbl 1207.54036
[12] Bartoszyński, T.; Judah, H.; Theory, Set, On the Structure of the Real Line (1995), A K Peters: A K Peters Wellesley, MA
[13] Bella, A., More on sequential properties of \(2^{\omega_1}\), Questions Answers Gen. Topology, 22, 1-4 (2004) · Zbl 1056.54005
[14] Bella, A.; Bonanzinga, M.; Matveev, M.; Tkachuk, V., Selective separability: General facts and behavior in countable spaces, Topology Proc., 32, 15-32 (2008) · Zbl 1165.54008
[15] Bella, A.; Bonanzinga, M.; Matveev, M., Variations of selective separability, Topology Appl., 156, 1241-1252 (2009) · Zbl 1168.54009
[16] Bella, A.; Bonanzinga, M.; Matveev, M., Addendum to “Variations of selective separability”, Topology Appl.. Topology Appl., Topology Appl., 157, 7, 2389-2391 (2010) · Zbl 1197.54038
[17] Bella, A.; Malykhin, V. I., Tightness and resolvability, Comment. Math. Univ. Carolin., 39, 177-184 (1998) · Zbl 0936.54004
[18] Bella, A.; Pavlov, O. I., Embeddings into pseudocompact spaces of countable tightness, Topology Appl., 138, 161-166 (2004) · Zbl 1041.54003
[19] Bella, A.; Simon, P., Pseudocompact Whyburn spaces of countable tightness need not be Fréchet, Proceedings of the 20th Summer Conference on Topology and its Applications. Proceedings of the 20th Summer Conference on Topology and its Applications, Topology Proc., 30, 2, 423-430 (2006) · Zbl 1136.54019
[20] Bella, A.; Tironi, G., Pseudoradial spaces, (Hart, K. P.; Nagata, J.; Vaughan, J. E., Encyclopedia of General Topology (2004), Elsevier Sci. Pub.), 165-168
[21] Bella, A.; Yaschenko, I. V., On AP and WAP spaces, Comment. Math. Univ. Carolin., 40, 531-536 (1999) · Zbl 1010.54040
[22] van Douwen, E. K., Applications of maximal topologies, Topology Appl., 51, 125-139 (1993) · Zbl 0845.54028
[23] Dow, A., On compact separable radial spaces, Canad. Math. Bull., 40, 422-432 (1997) · Zbl 0926.54014
[24] Dow, A.; Tkachenko, M.; Tkachuk, V.; Wilson, R., Topologies generated by discrete subspaces, Glas. Mat., 37, 57, 189-212 (2002) · Zbl 1009.54005
[25] Eisworth, T., On D-spaces, (Pearl, Elliott, Open Problems in Topology II, Chapter 1 (2007), Elsevier Publishing: Elsevier Publishing Amsterdam, The Netherlands), 129-134
[26] Engelking, R., General Topology (1989), Heldermann-Verlag: Heldermann-Verlag Berlin · Zbl 0684.54001
[27] Frankiewicz, R.; Shelah, S.; Zbierski, P., On closed P-sets with ccc in the space \(\omega^\ast \), J. Symbolic Logic, 58, 1171-1176 (1993) · Zbl 0794.03070
[28] Gartside, P. M., Cardinal invariants of monotonically normal spaces, Topology Appl., 77, 303-314 (1997) · Zbl 0872.54005
[29] Gruenhage, G., Generalized metric spaces, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), Elsevier), 423-501
[30] Gruenhage, G.; Natkaniec, T.; Piotrowski, Z., On thin, very thin, and slim dense sets, Topology Appl., 154, 817-833 (2007) · Zbl 1115.54004
[31] Gruenhage, G.; Sakai, M., Selective separability and its variations, Topology Appl., 158, 1352-1359 (2011) · Zbl 1228.54028
[32] S. Garcia-Ferreira, M. Hrušak, preprint.; S. Garcia-Ferreira, M. Hrušak, preprint.
[33] Juhász, I., Cardinal Function in Topology - Ten Years Later, Math. Centre Tracts, vol. 123 (1980), Mathematisch Centrum: Mathematisch Centrum Amsterdam · Zbl 0479.54001
[34] Jech, T., Set Theory: The Third Millennium Edition, revised and expanded (2002), Springer: Springer Berlin
[35] Juhász, I.; Shelah, S., \( \pi(X) = \delta(X)\) for compact \(X\), Topology Appl., 32, 289-294 (1989) · Zbl 0688.54002
[36] Juhász, I.; Soukup, L.; Szentmiklóssy, Z., \(D\)-forced spaces: A new approach to resolvability, Topology Appl., 156, 1800-1824 (2006) · Zbl 1101.54038
[37] Juhász, I.; Soukup, L.; Szentmiklóssy, Z., Resolvability and monotone normality, Israel J. Math., 166, 1-16 (2008) · Zbl 1155.54006
[38] Juhász, I.; Szentmiklóssy, Z., On d-separability of powers and \(C_p(X)\), Topology Appl., 155, 277-281 (2008) · Zbl 1134.54002
[39] Kunen, K.; Szymanski, A.; Tall, F., Baire irresolvable spaces and ideal theory, Ann. Math. Sil., 2, 98-107 (1986) · Zbl 0613.54018
[40] Kurepa, G., Le probléme de Souslin et les espaces abstrait, C. R. Acad. Sci. Paris, 204, 1049-1052 (1937) · JFM 62.0688.04
[41] Levy, R.; Porter, J., On two questions of Arhangelskii and Collins regarding submaximal spaces, Topology Proc., 21, 143-154 (1996) · Zbl 0891.54015
[42] Di Maio, G.; Kočinac, Lj.; Meccariello, E., Selection principles and hyperspace topologies, Topology Appl., 153, 912-923 (2005) · Zbl 1087.54007
[43] Malykhin, V. I., Extremally disconnected and nearly extremally disconnected groups, Dokl. Akad. Nauk SSSR, 220, 27-30 (1975) · Zbl 0322.22003
[44] Malykhin, V. I., On the resolvability of the product of two spaces and a problem of Katětov, Dokl. Akad. Nauk SSSR, 222, 725-729 (1975) · Zbl 0325.54017
[45] Moore, J., An \(L\)-space with a \(d\)-separable square, Topology Appl., 155, 4, 304-307 (2008) · Zbl 1146.54015
[46] Pelant, J.; Tkachenko, M. G.; Tkachuk, V. V.; Wilson, R. G., Pseudocompact Whyburn spaces need not be Fréchet, Proc. Amer. Math. Soc., 131, 10, 3257-3265 (2003) · Zbl 1028.54004
[47] Protasov, I. V., Filters and topologies on semigroups, Mat. Stud., 3, 120, 15-28 (1994) · Zbl 0927.22009
[48] Repovš, D.; Zdomskyy, L., On M-separability of countable spaces and function spaces, Topology Appl., 157, 2538-2541 (2010) · Zbl 1226.54029
[49] Sakai, M., Property \(C''\) and function spaces, Proc. Amer. Math. Soc., 104, 917-919 (1988) · Zbl 0691.54007
[50] Scheepers, M., Combinatorics of open covers. VI. Selectors for sequences of dense sets, Quaest. Math., 22, 1, 109-130 (1999) · Zbl 0972.91026
[51] Scheepers, M., Selection principles and covering properties in topology, Note Mat., 22, 2, 3-41 (2003/2004) · Zbl 1195.37029
[52] Schröder, J., Some answers concerning submaximal spaces, Questions Answers Gen. Topology, 17, 221-225 (1999) · Zbl 0936.54002
[53] Shapirovskii, B. E., Cardinal invariants in bicompacta, (Seminar on General Topology (1981), Moscow State Univ. Publ.: Moscow State Univ. Publ. Moscow), 162-187 · Zbl 0437.54008
[54] Simon, P., A note on cardinal invariants of square, Comment. Math. Univ. Carolin., 14, 2, 205-213 (1973) · Zbl 0258.54003
[55] Tsaban, B., Selection principles in mathematics: A milestone of open problems, Note Mat., 22, 2, 179-208 (2003/2004) · Zbl 1223.37059
[56] Tsaban, B., Some new directions in infinite-combinatorial topology, (Set Theory. Set Theory, Trends Math. (2006), Birkhäuser: Birkhäuser Basel), 225-255 · Zbl 1113.54002
[57] Tkachuk, V. V., Function spaces and d-separability, Quaest. Math., 28, 409-424 (2005) · Zbl 1091.54008
[58] Tkachuk, V. V.; Yaschenko, I. V., Almost closed sets and topologies they determine, Comment. Math. Univ. Carolin., 42, 2, 395-405 (2001) · Zbl 1053.54004
[59] Vaughan, J. E., Countably compact, locally countable \(T_2\)-spaces, Proc. Amer. Math. Soc., 80, 147-153 (1980) · Zbl 0444.54013
[60] Vaughan, J. E., Two spaces homeomorphic to Seq(p), Comment. Math. Univ. Carolin., 42, 209-218 (2001) · Zbl 1053.54033
[61] Williams, S.; Zhou, H., Order-like structure of monotonically normal spaces, Comment. Math. Univ. Carolin., 39, 207-217 (1998) · Zbl 0937.54012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.