Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$D$-spaces and finite unions


Author: Alexander Arhangel'skii
Translated by:
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
Published electronically: February 9, 2004
MathSciNet review: 2053991
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A. V. Arhangel'skii, $D$-spaces and covering properties. Submitted, 2002.
  • 2. A. V. Arhangel'skii and R. Z. Buzyakova, Addition theorems and $D$-spaces. Comment. Math. Univ. Carolinae 43,4 (2002), 653-663.
  • 3. C. R. Borges and A. C. Wehrly, A study of $D$-spaces. Topology Proc. 16 (1991), 7-15. MR 94a:54059
  • 4. D. K. Burke, A note on R. H. Bing's example G. In: Proc. V.P.I. Topology Conference, Lecture Notes in Math., vol. 375, Springer-Verlag, Berlin, 1974, pp. 47-52. MR 51:11426
  • 5. D. K. Burke, Covering properties. In: K. Kunen and J. Vaughan, Eds., Handbook of Set-theoretic Topology, Chapter 9, 347-422. North-Holland, Amsterdam, New York, Oxford, 1984. MR 86e:54030
  • 6. R. Z. Buzyakova, On $D$-property of strong $\Sigma $-spaces. Comment. Math. Universitatis Carolinae 43 (2002), 493-495. MR 2003j:54021
  • 7. J. Chaber, Metacompactness and the class MOBI. Fund. Math. 91 (1976), 211-217. MR 54:3646
  • 8. E. K. van Douwen and W. F. Pfeffer, Some properties of the Sorgenfrey line and related spaces. Pacific J. Math. 81:2 (1979), 371-377. MR 80h:54027
  • 9. E. K. van Douwen and H. H. Wicke, A real, weird topology on reals. Houston Journal of Mathematics 13:1 (1977), 141-152. MR 55:6390
  • 10. R. Engelking, General Topology. Polish Scientific Publishers, Warsaw, 1977. MR 58:18316b
  • 11. M. Ismail and A. Szymanski, On locally compact Hausdorff spaces with finite metrizability number. Topology Appl. 114:3 (2001), 285-293. MR 2002f:54003
  • 12. E. Michael and M. E. Rudin, Another note on Eberlein compacts. Pacific J. Math. 72 (1977), 497-499. MR 57:17584b
  • 13. A. J. Ostaszewski, Compact $\sigma $-metric spaces are sequential. Proc. Amer. Math. Soc. 68 (1978), 339-343. MR 57:7532
  • 14. M. E. Rudin, Dowker spaces. In: K. Kunen and J. Vaughan, Eds., Handbook of Set-theoretic Topology, Chapter 17, 761-780. North-Holland, Amsterdam, New York, Oxford, 1984. MR 86c:54018
  • 15. H. H. Wicke and J. M. Worrell, Jr., Point-countability and compactness. Proc. Amer. Math. Soc. 55 (1976), 427-431. MR 53:4001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54D20, 54F99

Retrieve articles in all journals with MSC (2000): 54D20, 54F99


Additional Information

Alexander Arhangel'skii
Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701
Email: arhangel@math.ohiou.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07336-8
Keywords: $D$-space, point-countable base, extent, subparacompact space, Lindel\"of degree, $aD$-space
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
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society