Researches on point-discrete families

Abstract

Based on the modern development of Metrization theorem for context, the main results obtained in recent ten years on point-discrete families are summarized. This paper mainly introduces the theory of the spaces with \sigma-point-discrete bases, the spaces with certain \sigma-point-discrete networks, and the relationship between the above spaces and the spaces with certain \sigma-compact-finite networks.

Graphical abstract

Keywords

Point-discrete families / compact-finite families / generalized metrizable spaces / k-net / weak bases

References

[1] AleksandrovP SFedorchukV VZcevV I. The main aspects in the development of set-theoretic topology. Uspekhi Mat Nauk, 1978, 33(3): 3−48 (in Russian)

[2] Arhangel’skii A V. An addition theorem for the weight of sets lying in bicompacts. Dokl Akad Nauk SSSR 1959; 126(2): 239–241

[3] Arhangel’skii A V. Mappings and spaces. Russian Math Surveys 1966; 21(4): 115–162

[4] AullC ELowenR. Handbook of the History of General Topology, Vol 1. Dordrecht: Kluwer Academic Publishers, 1997

[5] Bing R H. Metrization of topological spaces. Canad J Math 1951; 3(2): 175–186

[6] Boone J R. Some characterizations of paracompactness in k-spaces. Fund Math 1971; 72(2): 145–153

[7] Burke D K, Davis S W. Spaces with σ-weakly hereditarily closure preserving base. Topology Proc 2010; 35: 9–18

[8] Burke D K, Engelking R, Lutzer D J. Hereditarily closure-preserving collections and metrization. Proc Amer Math Soc 1975; 51(2): 483–488

[9] BurkeD KLutzerD J. Recent advances in the theory of generalized metric spaces. In: Topology Proc 9th Ann Spring Topological Conf (Memphis State Univ, 1975). Lect Notes Pure and Appl Math 24. New York: Marcel Dekker Inc, 1976, 1–70

[10] FangJ LHausdorff. Biographies of world-famous mathematicians. Beijing: Science Press, 1995, 1240–1246

[11] Franklin S P. Spaces in which sequences suffice. Fund Math 1965; 57(1): 107–115

[12] Gao Z. ℵ-space is invariant under perfect mappings. Questions Answers Gen Topology 1987; 5(2): 271–279

[13] Ge X, Shen J, Ge Y. Spaces with σ-weakly hereditarily closure-preserving sn-networks. Novi Sad J Math 2007; 37(1): 33–37

[14] Ge Y. On sn-Metrizable spaces. Acta Math Sin Engl Ser 2002; 45(2): 355–360

[15] JiangJ G. On the metrizability of affine compactness and topological spaces, Acta Math

[16] GruenhageG. Generalized metric spaces. In: Handbook of Set-theoretic Topology (Kunen K, Vaughan J E, eds). Amsterdam: Elsevier Science Publishers B V, 1984, 423–501

[17] GruenhageG. Generalized metric spaces and metrization. In: Recent Progress in General Topology (Hušek M, Van Mill J, eds). Papers from the Prague Topological Symp (Prague, 1991). Amsterdam: Elsevier Science Publishers B V, 1992, 239–274

[18] GruenhageG. Metrizable spaces and generalizations. In: Recent Progress in General Topology II (Hušek M, Van Mill J, eds). Papers from the Prague Topological Symp (Prague, 2001). Amsterdam: Elsevier Science Publishers B V, 2002, 201–225

[19] Gruenhage G. A survey of D-spaces. Contemp Math 2011; 533: 13–28

[20] GruenhageG. Generalized metrizable spaces. In: Recent Progress in General Topology III (Hart K P, Van Mill J, Simon P, eds). Papers from the Prague Topological Symp (Prague, 2011). Amsterdam: Atlantis Press, 2014, 471–505

[21] Guthrie J A. A characterization of ℵ0-spaces. General Topology Appl 1971; 1(2): 105–110

[22] HartK PVan MillJSimonP. Recent Progress in General Topology III. Papers from the Prague Topological Symp (Prague, 2011). Amsterdam: Atlantis Press, 2014

[23] HartK PNagataJVaughanJ E. Encyclopedia of General Topology. Amsterdam: Elsevier Science Publishers B V, 2004

[24] HausdorffF. Grundzüge der Mengenlehre. Leipzig: Veriag von Veit und Comp, 1914 (in German)

[25] Lašnev N. Closed images of metric spaces. Dokl Akad Nauk SSSR 1966; 170: 505–507

[26] Lin F C. σ–point–discrete cs*–networks and wcs*–networks. J Adv Research Pure Math 2010; 2(3): 7–12

[27] LinF C. Topological algebra and generalized metric spaces. Xiamen: Xiamen Univ Press, 2012

[28] Lin F C, Lin S. Some notes on sequence-covering maps. J Math Res Appl 2014; 34(1): 97–104

[29] Lin F C, Lin S. Sequence-covering maps on generalized metric spaces. Houston J Math 2014; 40(3): 927–943

[30] Lin F C, Shen R X. Some notes on σ-point-discrete sn-networks. Adv Math (China) 2010; 39(2): 212–216

[31] Lin S. On normal separable ℵ-spaces. Questions Answers Gen Topology 1987; 5: 249–254

[32] Lin S. A study of pseudobases. Questions Answers Gen Topology 1988; 6: 81–97

[33] Lin S. On sequence-covering s-mappings. Adv Math (China) 1996; 25(6): 548–551

[34] Lin S. A note on the Arens’ space and sequential fan. Topology Appl 1997; 81(3): 185–196

[35] LinS. Point countable coverings and sequential coverings mappings. Beijing: Science Press, 2002

[36] LinS. Generalized metric spaces and mappings. Beijing: Science Press, 2007

[37] Lin S, Tanaka Y. Point-countable k-networks, closed maps, and related results. Topology Appl 1994; 59(1): 79–86

[38] Lin Y, Yan L. A note on spaces with a σ-compact-finite weak base. Tsukuba J Math 2004; 28(1): 85–91

[39] Liu C. Spaces with a σ-compact finite k-network. Questions Answers Gen Topology 1992; 10: 81–87

[40] Liu C. Notes on g-metrizable spaces. Topology Proc 2005; 29(1): 207–215

[41] Liu C. On weak bases. Topology Appl 2005; 150(13): 91–99

[42] Liu C, Lin S. On countable-to-one maps. Topology Appl 2007; 154(2): 449–454

[43] Liu C, Lin S, Ludwig L D. Spaces with a σ-point-discrete weak base. Tsukuba J Math 2008; 32(1): 165–177

[44] Liu C, Ludwig L D. Nagata-Smirnov revisited: spaces with σ-WHCP bases. Topology Proc 2005; 29(2): 559–565

[45] Michael E A. Another note on paracompact spaces. Proc Amer Math Soc 1957; 8(4): 822–828

[46] Michael E A. A quintuple quotient quest. General Topology Appl 1972; 2(2): 91–138

[47] MoritaKNagataJ. Topics in General Topology. Amsterdam: Elsevier Science Publishers B V, 1989

[48] Nagata J. On a necessary and sufficient condition of metrizability. J Inst Polytech Osaka City Univ 1950; 1(2A): 93–100

[49] Okuyama A. On a generalization of Σ-spaces. Pacific J Math 1972; 42(2): 485–495

[50] O’Meara P. On paracompactness in function spaces with the compact open topology. Proc Amer Math Soc 1971; 29(1): 183–189

[51] Shang Y, Wang S Z. The discussion of spaces with σ-WHCP or σ-compact-finite sets. J Capital Norm Univ Nat Sci Ed 2007; 28(6): 16–21

[52] Shen R, Lin S. Spaces with σ-point-discrete ℵ0-weak bases. Appl Math J Chinese Univ 2013; 28B(l): 116–126

[53] Sirois-Dumais R. Quasi- and weakly-quasi-firs seven-countable spaces. Topology Appl 1980; 11(2): 223–230

[54] Smirnov M. On the metrization of topological spaces. Uspekhi Mat Nauk 1951; 6(6): 100–111

[55] Urysohn P. Zurn Metrisationsproblem. Math Ann 1925; 94(1): 309–315

[56] Van DouwenE K. Simultaneous extension of continuous functions. PhD Thesis. Amsterdam: Free Univ of Amsterdam, 1975

[57] EditorialCommittee of the Encyclopedia of China. Encyclopedia of China (Mathematics Vol). Beijing: Encyclopedia of China Press, 1988

RIGHTS & PERMISSIONS

2023 Higher Education Press 2023