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Keywords:
$k$-systems; $k$-networks; $k$-covers; $k$-spaces; point-countable families; hereditarily closure-preserving families
Summary:
The concepts of $k$-systems, $k$-networks and $k$-covers were defined by A. Arhangel’skiǐ in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among $k$-systems, $k$-networks and $k$-covers are further discussed and are established by $mk$-systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of $mk$-systems.
References:
[1] A.  Arhangel’skiǐ: On quotient mappings of metric spaces. Dokl. Akad. Nauk. SSSR 155 (1964), 247–250. (Russian)
[2] D. K.  Burke: ARRAY(0x9a06458). K. Kunen, J. E.  Vaughan (eds.), North-Holland, , 1984, pp. 347–422. MR 0776619
[3] J.  Chaber: Generalizations of Lašnev’s theorem. Fund. Math. 119 (1983), 85–91. MR 0731811 | Zbl 0547.54009
[4] Huaipeng Chen: On $s$-images of metric spaces. Topology Proc. 24 (1999), 95–103. MR 1802679
[5] L.  Foged: A characterization of closed images of metric spaces. Proc. Amer. Math. Soc. 95 (1985), 487–490. DOI 10.1090/S0002-9939-1985-0806093-3 | MR 0806093 | Zbl 0592.54027
[6] G.  Gruenhage: Generalized metric spaces. In: Handbook of Set-theoretic Topology, K. Kunen, J. E.  Vaughan (eds.), North-Holland, , 1984, pp. 423–501. MR 0776629 | Zbl 0555.54015
[7] G.  Gruenhage, E.  Michael, and Y.  Tanaka: Spaces determined by point-countable covers. Pacific J.  Math. 113 (1984), 303–332. DOI 10.2140/pjm.1984.113.303 | MR 0749538
[8] Jinjin  Li: $k$-covers and certain quotient images of paracompact locally compact spaces. Acta Math. Hungar. 95 (2002), 281–286. DOI 10.1023/A:1015645107703 | MR 1909598
[9] Jinjin  Li, Shou  Lin: Spaces with compact-countable $k$-systems. Acta Math. Hungar. 93 (2001), 1–6. DOI 10.1023/A:1013807430701 | MR 1924664
[10] Zhaowen  Li, Jinjin  Li: On Michael-Nagami’s problem. Questions Answers in General Topology 12 (1994), 85–91.
[11] Shou  Lin: On spaces with a $k$-network consisting of compact subsets. Topology Proc. 20 (1995), 185–190. MR 1429180
[12] R. A.  McCoy, I.  Ntantu: Countability properties of function spaces with set-open topologies. Topology Proc. 10 (1985), 329–345. MR 0876902
[13] E.  Michael: A note on closed maps and compact sets. Israel J.  Math. 2 (1964), 173–176. DOI 10.1007/BF02759940 | MR 0177396 | Zbl 0136.19303
[14] P.  O’Meara: On paracompactness in function spaces with the compact-open topology. Proc. Amer. Math. Soc. 29 (1971), 183–189. DOI 10.1090/S0002-9939-1971-0276919-3 | MR 0276919
[15] M.  Sakai: On spaces with a point-countable compact $k$-network. Yokohama Math.  J. 48 (2000), 13–16. MR 1788830 | Zbl 0964.54023
[16] Y.  Tanaka: Closed images of locally compact spaces and Fréchet space. Topology Proc. 7 (1982), 279–292. MR 0715799
[17] Y.  Tanaka: Point-countable $k$-systems and products of $k$-spaces. Pacific J.  Math. 101 (1982), 199–208. DOI 10.2140/pjm.1982.101.199 | MR 0671852 | Zbl 0498.54023
[18] Y.  Tanaka: Theory of $k$-networks II. Questions Answers in General Topology 19 (2001), 27–46. MR 1815344 | Zbl 0970.54023
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