Previous |  Up |  Next

Article

Keywords:
weak-open mappings; $\pi$-mappings; $g$-developable spaces; Cauchy spaces; cs-covers; sn-covers; weak-developments; point-star networks
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.
References:
[1] A. V.  Arhangel’skii: Mappings and spaces. Russian Math. Surveys 21 (1996), 115–162. DOI 10.1070/RM1966v021n04ABEH004169 | MR 0227950
[2] V. I.  Ponomarev: Axioms of countability of continuous mappings. Bull. Pol. Acad. Math. 8 (1960), 127–133. MR 0116314
[3] F.  Siwiec: On defining a space by a weak-base. Pacific J. Math. 52 (1974), 233–245. MR 0350706 | Zbl 0285.54022
[4] S.  Xia: Characterizations of certain $g$-first countable spaces. Adv. Math. 29 (2000), 61–64. MR 1769127 | Zbl 0999.54010
[5] Y.  Tanaka: Symmetric spaces, $g$-developable space and $g$-metrizable spaces. Math. Japonica 36 (1991), 71–84. MR 1093356
[6] P. S.  Alexandroff, V.  Niemytzki: The condition of metrizability of topological spaces and the axiom of symmetry. Mat. Sb. 3 (1938), 663–672.
[7] K. B.  Lee: On certain $g$-first countable spaces. Pacific J.  Math. 65 (1976), 113–118. DOI 10.2140/pjm.1976.65.113 | MR 0423307 | Zbl 0359.54022
[8] S. P.  Franklin: Spaces in which sequences suffice. Fund. Math. 57 (1965), 107–115. MR 0180954 | Zbl 0132.17802
[9] S.  Lin: Generalized Metric Spaces and Mappings. Chinese Scientific Publ., Beijing, 1995.
[10] S.  Lin: On sequence-covering $s$-mappings. Adv. Math. 25 (1996), 548–551. MR 1453163 | Zbl 0864.54026
[11] S.  Lin, Y.  Zhou, and P.  Yan: On sequence-covering $\pi $-mappings. Acta Math. Sinica 45 (2002), 1157–1164. MR 1959486
[12] R. W.  Heath: On open mappings and certain spaces satisfying the first countability axiom. Fund. Math. 57 (1965), 91–96. MR 0179763 | Zbl 0134.41802
[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
[14] D. K.  Burke: Cauchy sequences in semimetric spaces. Proc. Amer. Math. Soc. 33 (1972), 161–164. DOI 10.1090/S0002-9939-1972-0290328-3 | MR 0290328 | Zbl 0233.54015
[15] Y.  Ikeda, C.  Liu, and Y.  Tanaka: Quotient compact images of metric spaces, and related matters. Topology Appl. 122 (2002), 237–252. DOI 10.1016/S0166-8641(01)00145-6 | MR 1919303
[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. MR 0776629 | Zbl 0555.54015
[17] Y.  Tanaka, Z.  Li: Certain covering-maps and $k$-networks, and related matters. Topology Proc. 27 (2003), 317–334. MR 2048941 | Zbl 1075.54010
[18] Z.  Li: On the weak-open images of metric spaces. Czechoslovak Math.  J. 54 (2004), 393–400. DOI 10.1023/B:CMAJ.0000042377.80659.fb | MR 2059259 | Zbl 1080.54509
Partner of
EuDML logo