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)



On the Menger covering property and $ D$-spaces

Authors: Dušan Repovš and Lyubomyr Zdomskyy
Journal: Proc. Amer. Math. Soc. 140 (2012), 1069-1074
MSC (2010): Primary 54D20, 54A35; Secondary 54H05, 03E17
Published electronically: July 6, 2011
MathSciNet review: 2869091
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main results of this paper are:

  • It is consistent that every subparacompact space $ X$ of size $ \omega_1$ is a $ D$-space.
  • If there exists a Michael space, then all productively Lindelöf spaces have the Menger property and, therefore, are $ D$-spaces.
  • Every locally $ D$-space which admits a $ \sigma$-locally finite cover by Lindelöf spaces is a $ D$-space.

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

  • 1. Alas, O.; Aurichi, L.F.; Junqueira, L.R.; Tall, F.D., Non-productively Lindelöf spaces and small cardinals, Houston J. Math., to appear.
  • 2. Aurichi, L.F., $ D$-spaces, topological games, and selection principles, Topology Proc. 36 (2010), 107-122. MR 2591778
  • 3. Borges, C.R.; Wehrly, A.C., A study of $ D$-spaces, Topology Proc. 16 (1991), 7-15. MR 1206448 (94a:54059)
  • 4. Bukovský, L.; Haleš, J., On Hurewicz properties, Topology Appl. 132 (2003), 71-79. MR 1990080 (2004f:03085)
  • 5. Burke, D.K., Covering properties, in: Handbook of Set-Theoretic Topology (K. Kunen, J.E. Vaughan, eds.), North Holland, Amsterdam, 1984, 347-422. MR 776628 (86e:54030)
  • 6. Van Douwen, E.K.; Pfeffer, W.F., Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (1979), 371-377. MR 547605 (80h:54027)
  • 7. Gruenhage, G., A survey on $ D$ spaces, Contemp. Math., Vol. 533, Amer. Math. Soc., Providence, RI, 2011.
  • 8. Hurewicz, W., Über die Verallgemeinerung des Borellschen Theorems, Math. Z. 24 (1925), 401-421.
  • 9. Just, W.; Miller, A.W.; Scheepers, M.; Szeptycki, P.J., The combinatorics of open covers. II, Topology Appl. 73 (1996), 241-266. MR 1419798 (98g:03115a)
  • 10. Martínez, J.C.; Soukup, L., The D-property in unions of scattered spaces, Topology Appl. 156 (2009), 3086-3090. MR 2556068 (2010k:54027)
  • 11. Moore, J.T., Some of the combinatorics related to Michael's problem, Proc. Amer. Math. Soc. 127 (1999), 2459-2467. MR 1486743 (99j:54008)
  • 12. Repický, M., Another proof of Hurewicz theorem, Tatra Mt. Math. Publ., to appear.
  • 13. Repovš, D.; Semenov, P., Continuous selections of multivalued mappings. Mathematics and its Applications, 455. Kluwer Academic Publishers, Dordrecht, 1998. MR 1659914 (2000a:54002)
  • 14. Scheepers, M., Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), 31-62. MR 1378387 (97h:90123)
  • 15. Shi, W.; Zhang, H., A note on $ D$-spaces, preprint, 2010.
  • 16. Tall, F.D., Productively Lindelöf spaces may all be $ D$, Canad. Math. Bulletin, to appear.
  • 17. Tall, F.D., Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or $ D$, Topology Appl., to appear.
  • 18. Tall, F.D., A note on productively Lindelöf spaces, preprint, 2010.
  • 19. Tall, F.D., Set-theoretic problems concerning Lindelöf spaces, preprint, 2010.
  • 20. Telgársky, R., Topological games: On the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (1987), 227-276. MR 892457 (88d:54046)
  • 21. Tsaban, B., Selection principles and special sets of reals, in: Open problems in topology. II (edited By Elliott Pearl), Elsevier Sci. Publ., 2007, pp. 91-108.
  • 22. Vaughan, J., Small uncountable cardinals and topology, in: Open problems in topology (J. van Mill, G.M. Reed, eds.), Elsevier Sci. Publ., 1990, pp. 195-218. MR 1078647
  • 23. Zdomskyy, L., A semifilter approach to selection principles, Comment. Math. Univ. Carolin. 46 (2005), 525-539. MR 2174530 (2006g:54028)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 54D20, 54A35, 54H05, 03E17

Retrieve articles in all journals with MSC (2010): 54D20, 54A35, 54H05, 03E17

Additional Information

Dušan Repovš
Affiliation: Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001

Lyubomyr Zdomskyy
Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria

Keywords: $D$-space, subparacompactness, Menger property, (productively) Lindelöf space, Michael space
Received by editor(s): September 28, 2010
Received by editor(s) in revised form: December 4, 2010, and December 13, 2010
Published electronically: July 6, 2011
Additional Notes: The first author was supported by SRA grants P1-0292-0101 and J1-2057-0101.
The second author acknowledges the support of FWF grant P19898-N18
The authors would also like to thank Leandro Aurichi, Franklin Tall, and Hang Zhang for kindly making their recent papers available to us.
Communicated by: Julia Knight
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society