# Article

 Title: Spaces with countable $sn$-networks  (English) Author: Ying, Ge Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 45 Issue: 1 Year: 2004 Pages: 169-176 . Category: math . Summary: In this paper, we prove that a space $X$ is a sequentially-quotient $\pi$-image of a metric space if and only if $X$ has a point-star $sn$-network consisting of $cs^*$-covers. By this result, we prove that a space $X$ is a sequentially-quotient $\pi$-image of a separable metric space if and only if $X$ has a countable $sn$-network, if and only if $X$ is a sequentially-quotient compact image of a separable metric space; this answers a question raised by Shou Lin affirmatively. We also obtain some results on spaces with countable $sn$-networks. Keyword: separable metric space Keyword: sequentially-quotient ($\pi$ Keyword: compact) mapping Keyword: point-star $sn$-network Keyword: $cs*$-cover MSC: 54C05 MSC: 54C10 MSC: 54D65 MSC: 54E40 idZBL: Zbl 1098.54025 idMR: MR2076868 . Date available: 2009-05-05T16:44:06Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119445 . Reference: [1] Engelking R.: General Topology.Polish Scientific Publishers, Warszawa, 1977. Zbl 0684.54001, MR 0500780 Reference: [2] Foged L.: Characterizations of $\aleph$-spaces.Pacific J. Math. 110 (1984), 59-63. Zbl 0542.54030, MR 0722737 Reference: [3] Franklin S.P.: Spaces in which sequence suffice.Fund. Math. 57 (1965), 107-115. MR 0180954 Reference: [4] Ge Y.: On $sn$-metrizable spaces.Acta Math. Sinica 45 (2002), 355-360 (in Chinese). Zbl 1010.54027, MR 1928146 Reference: [5] Gruenhage G.: Generalized metric spaces.in: K. Kunen and J.E. Vaughan, Eds., Handbook of Set-Theoretic Topology, Amsterdam, North-Holland, pp.423-501. Zbl 0794.54034, MR 0776629 Reference: [6] Lin S.: On normal separable $\aleph$-space.Questions Answers Gen. Topology 5 (1987), 249-254. MR 0917881 Reference: [7] Lin S.: A survey of the theory of $\aleph$-space.Questions Answers Gen. Topology 8 (1990), 405-419. MR 1065288 Reference: [8] Lin S.: Generalized Metric Spaces and Mappings.Chinese Science Press, Beijing, 1995 (in Chinese). MR 1375020 Reference: [9] Lin S.: A note on the Arens' spaces and sequential fan.Topology Appl. 81 (1997), 185-196. MR 1485766 Reference: [10] Lin S., Yan P.: Sequence-covering maps of metric spaces.Topology Appl. 109 (2001), 301-314. Zbl 0966.54012, MR 1807392 Reference: [11] Lin S., Yan P.: On sequence-covering compact mappings.Acta Math. Sinica 44 (2001), 175-182 (in Chinese). Zbl 1005.54031, MR 1819992 Reference: [12] Lin S.: Point-star networks and $\pi$-mappings.Selected Papers of Chinese Topology Symposium, Chinese Fuzhou Teachers University Publ., Chinese Fuzhou, 2001 (in Chinese). Reference: [13] Michael E.A.: $\aleph_0$-spaces.J. Math. Mech. 15 (1966), 983-1002. MR 0206907 Reference: [14] O'Meara P.: A new class of topological spaces.Univ. of Alberta Dissertation, 1966. Reference: [15] O'Meara P.: On paracompactness in function spaces with the compact-open topology.Proc. Amer. Math. Soc. 29 (1971), 183-189. Zbl 0214.21105, MR 0276919 Reference: [16] Ponomarev V.I.: Axioms of countability and continuous mappings.Bull. Pol. Acad. Math. 8 (1960), 127-133 (in Russian). MR 0116314 Reference: [17] Tanaka Y.: $\sigma$-Hereditarily closure-preserving $k$-networks and $g$-metrizability.Proc. Amer. Math. Soc. 112 (1991), 283-290. Zbl 0770.54031, MR 1049850 Reference: [18] Tanaka Y.: Theory of $k$-networks.Questions Answers Gen. Topology 12 (1994), 139-164. Zbl 0833.54015, MR 1288740 .

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