×

Resolvability: A selective survey and some new results. (English) Zbl 0866.54004

Summary: The authors give a brief introduction to the theory of spaces which are resolvable in the sense introduced by E. Hewitt [Duke Math. J. 10, 309-333 (1943; Zbl 0060.39407)]. The new results presented here are:
(A) A countably compact regular Hausdorff space without isolated points is \(\omega\)-resolvable – that is, it admits an infinite family of pairwise disjoint dense subsets.
(B) Among Tikhonov topologies without isolated points on a fixed set, no pseudocompact topology is maximal.

MSC:

54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54-02 Research exposition (monographs, survey articles) pertaining to general topology
54C05 Continuous maps
54G10 \(P\)-spaces
22A05 Structure of general topological groups
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54H11 Topological groups (topological aspects)

Citations:

Zbl 0060.39407
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, D. R., On connected irresolvable Hausdorff spaces, (Proc. Amer. Math. Soc., 16 (1965)), 463-466 · Zbl 0127.13003
[2] Arhangel’skiĭ, A. V.; Franklin, S. P., Ordinal invariants for topological spaces, Michigan Math. J., 15, 506 (1968) · Zbl 0167.51102
[3] Bernstein, A. R., A new kind of compactness for topological spaces, Fund. Math., 66, 185-193 (1970) · Zbl 0198.55401
[4] Bešlagić, A.; Levy, R., Irresolvable products, (Proc. Summer Conference on General Topology and Applications. Proc. Summer Conference on General Topology and Applications, 1995 (1996), Ann. New York Acad. Sci.: Ann. New York Acad. Sci. Oxford), to appear · Zbl 0895.54004
[5] Cameron, D. E., Maximal and minimal topologies, Trans. Amer. Math. Soc., 160, 229-248 (1971) · Zbl 0202.22302
[6] Cameron, D. E., A class of maximal topologies, Pacific J. Math., 70, 101-104 (1977) · Zbl 0335.54022
[7] Ceder, J. G., On maximally resolvable spaces, Fund. Math., 55, 87-93 (1964) · Zbl 0139.40401
[8] Ceder, J.; Pearson, T., On products of maximally resolvable spaces, Pacific J. Math., 22, 31-45 (1967) · Zbl 0153.24201
[9] Comfort, W. W.; Feng, Li, The union of resolvable spaces is resolvable, Math. Japon., 38, 413-414 (1993) · Zbl 0769.54002
[10] Comfort, W. W.; Gladdines, H.; van Mill, J., Proper pseudocompact subgroups of pseudocompact Abelian groups, (Andima, S.; Itzkowitz, G.; Kong, T. Y.; Kopperman, R.; Misra, P. R.; Todd, A., Papers on General Topology and Applications, Proc. of Queens College Summer Conference on General Topology and Applications. Papers on General Topology and Applications, Proc. of Queens College Summer Conference on General Topology and Applications, June, 1992. Papers on General Topology and Applications, Proc. of Queens College Summer Conference on General Topology and Applications. Papers on General Topology and Applications, Proc. of Queens College Summer Conference on General Topology and Applications, June, 1992, Ann. New York Acad. Sci., 728 (1994)), 237-247, (New York) · Zbl 0915.54029
[11] Comfort, W. W.; Masaveu, O.; Zhou, Hao, Resolvability in topology and in topological groups, (Proc. Ninth Annual Conference on Topology and Applications. Proc. Ninth Annual Conference on Topology and Applications, June, 1993. Proc. Ninth Annual Conference on Topology and Applications. Proc. Ninth Annual Conference on Topology and Applications, June, 1993, Ann. New York Acad. Sci., 767 (1995)), 17-27, (New York) · Zbl 0919.54031
[12] Comfort, W. W.; Negrepontis, S., The Theory of Ultrafilters, (Grundl. Math. Wissensch., 221 (1974), Springer: Springer New York) · Zbl 0298.02004
[13] Comfort, W. W.; van Mill, J., Groups with only resolvable group topologies, (Proc. Amer. Math. Soc., 120 (1993)), 687-696 · Zbl 0820.22001
[14] van Douwen, E. K., Applications of maximal topologies, Topology Appl., 51, 125-139 (1993) · Zbl 0845.54028
[15] F.W. Eckertson, Resolvable, not maximally resolvable spaces, Manuscript submitted for publication.; F.W. Eckertson, Resolvable, not maximally resolvable spaces, Manuscript submitted for publication. · Zbl 0918.54035
[16] El’kin, A. G., Decomposition of spaces, Dokl. Akad. Nauk SSSR, 186, 1, 9-12 (1969) · Zbl 0202.53701
[17] El’kin, A. G., Dokl. Akad. Nauk SSSR, 186, 4, 765-768 (1969) · Zbl 0199.57302
[18] El’kin, A. G., Resolvable spaces which are not maximally resolvable, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 24, 4, 66-70 (1969) · Zbl 0183.51204
[19] El’kin, A. G., Ultrafilters and undecomposable spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 24, 5, 51-56 (1969) · Zbl 0196.26102
[20] Engelking, R., General Topology (1989), Heldermann: Heldermann Berlin · Zbl 0684.54001
[21] Feng, Li; Masaveu, O., Exactly \(n\)-resolvable spaces and \(ω\)-resolvability (1995), Manuscript submitted for publication · Zbl 0998.54026
[22] Frolík, Z., Types of ultrafilters on countable sets, (Novák, J., General Topology and Its Relations to Modern Analysis and Algebra II, Proc. Second Prague Topological Symposium (1966), Academia: Academia Berlin), 142-143
[23] Frolik, Z., Sums of ultrafilters, Bull. Amer. Math. Soc., 73, 87-91 (1967) · Zbl 0166.18602
[24] Ganster, M., Preopen sets and resolvable spaces, Kyungpook Math. J., 27, 135-143 (1987) · Zbl 0665.54001
[25] Hewitt, E., A problem of set-theoretic topology, Duke Math. J., 10, 309-333 (1943) · Zbl 0060.39407
[26] Illanes, A., Finite and \(ω\)-resolvability, (Proc. Amer. Math. Soc., 124 (1996)), 1243-1246 · Zbl 0856.54010
[27] Kunen, K., Maximal \(σ\)-independent families, Fund. Math., 117, 75-80 (1983) · Zbl 0532.03024
[28] Kunen, K.; Szymański, A.; Tall, F., Baire irresolvable spaces and ideal theory, Ann. Math. Sil., 2, 14, 98-107 (1986) · Zbl 0613.54018
[29] Malykhin, V. I., On resolvable and maximal spaces, Dokl. Akad. Nauk SSSR, 218 (1974) · Zbl 0306.54008
[30] Malykhin, V. I., Extremally disconnected and similar groups, Dokl. Akad. Nauk SSSR, 220, 27-30 (1975) · Zbl 0444.22001
[31] Malykhin, V. I.; Protasov, I. V., Maximal resolvability of bounded groups, Topology Appl. (1996), to appear · Zbl 0866.54032
[32] Masaveu, O., Dense subsets of some topological groups, (Ph.D. Thesis (1995), Wesleyan University: Wesleyan University Prague)
[33] von Neumann, J., Almost periodic functions in a group I, Trans. Amer. Math. Soc., 36, 445-492 (1934) · Zbl 0009.34902
[34] Padmavally, K., An example of a connected irresolvable Hausdorff space, Duke Math. J., 20, 513-520 (1953) · Zbl 0052.18904
[35] Porter, J. R.; Stephenson, R. M.; Woods, R. G., Maximal pseudocompact spaces, Comment. Math. Univ. Carolin., 35, 1, 127-145 (1994) · Zbl 0804.54004
[36] Protasov, I. V., Discrete subsets of topological groups, Mat. Zametki, 55, 1, 150-151 (1994) · Zbl 0836.22003
[37] Pytke’ev, E. G., On maximally resolvable spaces, Trudy Mat. Inst. Steklov, 154, 209-213 (1983) · Zbl 0529.54005
[38] Bhaskara Rao, K. P.S., On N-resolvability (August, 1994), Manuscript privately circulated
[39] Velichko, N. V., Mat. Zametki, 19, 1, 109-114 (1976) · Zbl 0809.46021
[40] Villegas-Silva, L. M., On resolvable spaces and groups, Comment. Math. Univ. Carolin., 36, 570-584 (1995) · Zbl 0837.22001
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.