Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON CS-STARCOMPACT SPACES

  • Song, Yan-Kui (Institute of Mathematics School of Mathematical Sciences Nanjing Normal University)
  • Received : 2010.10.11
  • Published : 2012.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

A space X is cs-starcompact if for every open cover $\mathcal{U}$ of X, there exists a convergent sequence S of X such that St(S, $\mathcal{U}$) = X, where $St(S,\mathcal{U})\;=\; \cup\{U{\in}\mathcal{U}:U{\cap}S{\neq}\phi\}$. In this paper, we prove the following statements: (1) There exists a Tychonoff cs-starcompact space having a regular-closed subset which is not cs-starcompact; (2) There exists a Hausdorff cs-starcompact space with arbitrary large extent; (3) Every Hausdorff centered-Lindel$\ddot{o}$f space can be embedded in a Hausdorff cs-starcompact space as a closed subspace.

Keywords

References

  1. M. Bonanzinga and M. V. Matveev, Closed subspaces of star-Lindelof and related spaces, East-West J. Math. 2 (2000), no. 2, 171-179.
  2. M. Bonanzinga and M. V. Matveev, Products of star-Lindelof and related spaces, Houston J. Math. 27 (2001), no. 1, 45-57.
  3. E. K. van Douwn, G. M. Reed, A. W. Roscoe, and I. J. Tree, Star covering properties, Topology Appl. 39 (1991), no. 1, 71-103. https://doi.org/10.1016/0166-8641(91)90077-Y
  4. R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag, Berlin, 1989.
  5. W. M. Fleischman, A new extension of countable compactness, Fund. Math. 67 (1970), 1-7. https://doi.org/10.4064/fm-67-1-1-9
  6. M. V. Matveev, A survey on star-covering properties, Topological Atlas, No. 330, 1998.
  7. M. V. Matveev, How weak is weak extent?, Topology Appl. 119 (2002), no. 2, 229-232. https://doi.org/10.1016/S0166-8641(01)00061-X
  8. J. van Mill, V. V. Tkachuk, and R. G Wilson, Classes defined by stars and neighbourhood assignments, Topology Appl. 154 (2007), no. 10, 2127-2134. https://doi.org/10.1016/j.topol.2006.03.029
  9. Y.-K. Song, On K-starcompact spaces, Bull. Malays. Math. Sci. Soc. (2) 30 (2007), no. 1, 59-64.
  10. Y.-K. Song, On countable K-covering properties, Appl. Gen. Topol. 8 (2007), no. 2, 249-258. https://doi.org/10.4995/agt.2007.1890