| Title: | On weak-open $\pi$-images of metric spaces (English) | 
| Author: | Li, Zhaowen | 
| Language: | English | 
| Journal: | Czechoslovak Mathematical Journal | 
| ISSN: | 0011-4642 (print) | 
| ISSN: | 1572-9141 (online) | 
| Volume: | 56 | 
| Issue: | 3 | 
| Year: | 2006 | 
| Pages: | 1011-1018 | 
| Summary lang: | English | 
| . | 
| Category: | math | 
| . | 
| Summary: | In this paper, we give some characterizations of metric spaces under weak-open $\pi$-mappings, which prove that a space is $g$-developable (or Cauchy) if and only if it is a weak-open $\pi$-image of a metric space. (English) | 
| Keyword: | weak-open mappings | 
| Keyword: | $\pi$-mappings | 
| Keyword: | $g$-developable spaces | 
| Keyword: | Cauchy spaces | 
| Keyword: | cs-covers | 
| Keyword: | sn-covers | 
| Keyword: | weak-developments | 
| Keyword: | point-star networks | 
| MSC: | 54C10 | 
| MSC: | 54D55 | 
| MSC: | 54E40 | 
| MSC: | 54E99 | 
| idZBL: | Zbl 1164.54365 | 
| idMR: | MR2261673 | 
| . | 
| Date available: | 2009-09-24T11:40:47Z | 
| Last updated: | 2020-07-03 | 
| Stable URL: | http://hdl.handle.net/10338.dmlcz/128126 | 
| . | 
| Reference: | [1] A. V.  Arhangel’skii: Mappings and spaces.Russian Math. Surveys 21 (1996), 115–162. MR 0227950, 10.1070/RM1966v021n04ABEH004169 | 
| Reference: | [2] V. I.  Ponomarev: Axioms of countability of continuous mappings.Bull. Pol. Acad. Math. 8 (1960), 127–133. MR 0116314 | 
| Reference: | [3] F.  Siwiec: On defining a space by a weak-base.Pacific J. Math. 52 (1974), 233–245. Zbl 0285.54022, MR 0350706 | 
| Reference: | [4] S.  Xia: Characterizations of certain $g$-first countable spaces.Adv. Math. 29 (2000), 61–64. Zbl 0999.54010, MR 1769127 | 
| Reference: | [5] Y.  Tanaka: Symmetric spaces, $g$-developable space and $g$-metrizable spaces.Math. Japonica 36 (1991), 71–84. MR 1093356 | 
| Reference: | [6] P. S.  Alexandroff, V.  Niemytzki: The condition of metrizability of topological spaces and the axiom of symmetry.Mat. Sb. 3 (1938), 663–672. | 
| Reference: | [7] K. B.  Lee: On certain $g$-first countable spaces.Pacific J.  Math. 65 (1976), 113–118. Zbl 0359.54022, MR 0423307, 10.2140/pjm.1976.65.113 | 
| Reference: | [8] S. P.  Franklin: Spaces in which sequences suffice.Fund. Math. 57 (1965), 107–115. Zbl 0132.17802, MR 0180954, 10.4064/fm-57-1-107-115 | 
| Reference: | [9] S.  Lin: Generalized Metric Spaces and Mappings.Chinese Scientific Publ., Beijing, 1995. | 
| Reference: | [10] S.  Lin: On sequence-covering $s$-mappings.Adv. Math. 25 (1996), 548–551. Zbl 0864.54026, MR 1453163 | 
| Reference: | [11] S.  Lin, Y.  Zhou, and P.  Yan: On sequence-covering $\pi $-mappings.Acta Math. Sinica 45 (2002), 1157–1164. MR 1959486 | 
| Reference: | [12] R. W.  Heath: On open mappings and certain spaces satisfying the first countability axiom.Fund. Math. 57 (1965), 91–96. Zbl 0134.41802, MR 0179763, 10.4064/fm-57-1-91-96 | 
| Reference: | [13] J. A.  Kofner: On a new class of spaces and some problems of symmetrizability theory.Soviet Math. Dokl. 10 (1969), 845–848. Zbl 0202.53702 | 
| Reference: | [14] D. K.  Burke: Cauchy sequences in semimetric spaces.Proc. Amer. Math. Soc. 33 (1972), 161–164. Zbl 0233.54015, MR 0290328, 10.1090/S0002-9939-1972-0290328-3 | 
| Reference: | [15] Y.  Ikeda, C.  Liu, and Y.  Tanaka: Quotient compact images of metric spaces, and related matters.Topology Appl. 122 (2002), 237–252. MR 1919303, 10.1016/S0166-8641(01)00145-6 | 
| Reference: | [16] G.  Gruenhage: Generalized metric spaces.In: Handbook of Set-theoretic Topology, K. Kunen, J. E. Vaughan (eds.), North-Holland, Amsterdam, 1984, pp. 423–501. Zbl 0555.54015, MR 0776629 | 
| Reference: | [17] Y.  Tanaka, Z.  Li: Certain covering-maps and $k$-networks, and related matters.Topology Proc. 27 (2003), 317–334. Zbl 1075.54010, MR 2048941 | 
| Reference: | [18] Z.  Li: On the weak-open images of metric spaces.Czechoslovak Math.  J. 54 (2004), 393–400. Zbl 1080.54509, MR 2059259, 10.1023/B:CMAJ.0000042377.80659.fb | 
| . |