# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
dense subspace; perfect space; Moore space; Čech-complete; $p$-space; $\sigma$-disjoint base; uniform base; pseudocompact; point-countable base; pseudo-$\omega _1$-compact
Summary:
We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu$ of dense metrizable subspaces, then $X$ is separable and metrizable.
References:
[1] Arhangel'skiĭ A.V.: Some metrization theorems. Uspekhi Mat. Nauk 18 (1963), no. 5, 139–145 (in Russian). MR 0156318
[2] Arhangel'skii A.V.: On a class of spaces containing all metric and all locally compact spaces. Mat. Sb. 67(109) (1965), 55–88; English translation: Amer. Math. Soc. Transl. 92 (1970), 1–39. MR 0190889
[3] Arhangel'skii A.V.: A generalization of Čech-complete spaces and Lindelöf $\Sigma$-spaces. Comment. Math. Univ. Carolin. 54 (2013), no. 2, 121–139. MR 3067699 | Zbl 1289.54085
[4] Arhangel'skii A.V., Choban M.M.: Spaces with sharp bases and with other special bases of countable order. Topology Appl. 159 (2012), no. 5, 1578-1590. DOI 10.1016/j.topol.2011.03.015 | MR 2891424 | Zbl 1245.54025
[5] Arhangel'skii A.V., Tokgöz S.: Paracompactness and remainders: around Henriksen-Isbell's Theorem. Questions Answers Gen. Topology 32 (2014), 5–15. MR 3222525 | Zbl 1305.54032
[6] van Douwen E.K., Tall F., Weiss W.: Non-metrizable hereditarily Lindelöf spaces with point-countable bases from CH. Proc. Amer. Math. Soc. 64 (1977), 139–145. MR 0514998 | Zbl 0356.54020
[7] Engelking R.: General Topology. PWN, Warszawa, 1977. MR 0500780 | Zbl 0684.54001
[8] Filippov V.V.: On feathered paracompacta. Dokl. Akad. Nauk SSSR 178 (1968), no. 3, 555–558. MR 0227935 | Zbl 0167.21103
[9] Gruenhage G.: Metrizable spaces and generalizations. in: M. Hušek and J. van Mill, Eds., Recent Progress in General Topology, II, North-Holland, Amsterdam, 2002, Chapter 8, pp. 203–221. MR 1969999 | Zbl 1029.54036
[10] Ismail M., Szymanski A.: On the metrizability number and related invariants of spaces. Topology Appl. 63 (1995), 69–77. DOI 10.1016/0166-8641(95)90009-8 | MR 1328620 | Zbl 0864.54001
[11] Ismail M., Szymanski A.: On the metrizability number and related invariants of spaces, II. Topology Appl. 71 (1996), 179–191. DOI 10.1016/0166-8641(95)00082-8 | MR 1399555 | Zbl 0864.54001
[12] Ismail M., Szymanski A.: On locally compact Hausdorff spaces with finite metrizability number. Topology Appl. 114 (2001), 285–293. DOI 10.1016/S0166-8641(00)00043-2 | MR 1838327 | Zbl 1012.54002
[13] Kuratowski K.: Topology, vol. 1. PWN, Warszawa, 1966.
[14] Michael E.A., Rudin M.E.: Another note on Eberlein compacta. Pacific J. Math. 72 (1977), no. 2, 497–499. DOI 10.2140/pjm.1977.72.497 | MR 0478093
[15] Oka S.: Dimension of finite unions of metric spaces. Math. Japon. 24 (1979), 351–362. MR 0557465 | Zbl 0429.54017

Partner of