$D$-spaces and finite unions
HTML articles powered by AMS MathViewer
- by Alexander Arhangel’skii PDF
- Proc. Amer. Math. Soc. 132 (2004), 2163-2170 Request permission
Abstract:
This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) “nice" subspaces? Our approach is based on the notion of a $D$-space introduced by E. van Douwen and on a generalization of this notion, the notion of $aD$-space. It is proved that if a space $X$ is the union of a finite family of subparacompact subspaces, then $X$ is an $aD$-space. Under $(CH)$, it follows that if a separable normal $T_1$-space $X$ is the union of a finite number of subparacompact subspaces, then $X$ is Lindelöf. It is also established that if a regular space $X$ is the union of a finite family of subspaces with a point-countable base, then $X$ is a $D$-space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.References
- A. V. Arhangel’skii, $D$-spaces and covering properties. Submitted, 2002.
- A. V. Arhangel’skii and R. Z. Buzyakova, Addition theorems and $D$-spaces. Comment. Math. Univ. Carolinae 43,4 (2002), 653-663.
- Carlos R. Borges and Albert C. Wehrly, A study of $D$-spaces, Topology Proc. 16 (1991), 7–15. MR 1206448
- Dennis K. Burke, A note on R. H. Bing’s Example $G$, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 47–52. MR 0375230
- Dennis K. Burke, Covering properties, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 347–422. MR 776628
- Raushan Z. Buzyakova, On $D$-property of strong $\Sigma$ spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 3, 493–495. MR 1920524
- J. Chaber, Metacompactness and the class MOBI, Fund. Math. 91 (1976), no. 3, 211–217. MR 415561, DOI 10.4064/fm-91-3-211-217
- Eric K. van Douwen and Washek F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (1979), no. 2, 371–377. MR 547605, DOI 10.2140/pjm.1979.81.371
- Eric K. van Douwen and Howard H. Wicke, A real, weird topology on the reals, Houston J. Math. 3 (1977), no. 1, 141–152. MR 433414
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- M. Ismail and A. Szymanski, On locally compact Hausdorff spaces with finite metrizability number, Topology Appl. 114 (2001), no. 3, 285–293. MR 1838327, DOI 10.1016/S0166-8641(00)00043-2
- E. Michael and M. E. Rudin, A note on Eberlein compacts, Pacific J. Math. 72 (1977), no. 2, 487–495. MR 478092, DOI 10.2140/pjm.1977.72.487
- A. J. Ostaszewski, Compact $\sigma$-metric Hausdorff spaces are sequential, Proc. Amer. Math. Soc. 68 (1978), no. 3, 339–343. MR 467677, DOI 10.1090/S0002-9939-1978-0467677-4
- Mary Ellen Rudin, Dowker spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 761–780. MR 776636
- H. H. Wicke and J. M. Worrell Jr., Point-countability and compactness, Proc. Amer. Math. Soc. 55 (1976), no. 2, 427–431. MR 400166, DOI 10.1090/S0002-9939-1976-0400166-X
Additional Information
- Alexander Arhangel’skii
- Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701
- Email: arhangel@math.ohiou.edu
- Received by editor(s): October 21, 2002
- Received by editor(s) in revised form: April 14, 2003
- Published electronically: February 9, 2004
- Communicated by: Alan Dow
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2163-2170
- MSC (2000): Primary 54D20; Secondary 54F99
- DOI: https://doi.org/10.1090/S0002-9939-04-07336-8
- MathSciNet review: 2053991