DOI QR코드

DOI QR Code

ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES

  • Hong, Woo-Chorl (DEPARTMENT OF MATHEMATICS EDUCATION PUSAN NATIONAL UNIVERSITY)
  • Published : 2007.04.30

Abstract

In this paper, we study on C-closed spaces, SC-closed spaces and related spaces. We show that a sequentially compact SC-closed space is sequential and as corollaries obtain that a sequentially compact space with unique sequential limits is sequential if and only if it is C-closed [7, 1.19 Proposition] and every sequentially compact SC-closed space is C-closed. We also show that a countably compact WAP and C-closed space is sequential and obtain that a countably compact (or compact or sequentially compact) WAP-space with unique sequential limits is sequential if and only if it is C-closed as a corollary. Finally we prove that a weakly discretely generated AP-space is C-closed. We then obtain that every countably compact (or compact or sequentially compact) weakly discretely generated AP-space is $Fr\acute{e}chet$-Urysohn with unique sequential limits, for weakly discretely generated AP-spaces, unique sequential limits ${\equiv}KC{\equiv}C-closed{\equiv}SC-closed$, and every continuous surjective function from a countably compact (or compact or sequentially compact) space onto a weakly discretely generated AP-space is closed as corollaries.

Keywords

References

  1. A. V. Arhangel'skii and L. S. Pontryagin(eds.), General Topology I, Encyclopaedia of Mathematical Sciences, vol. 17, Springer-Verlage, Berlin, 1990
  2. A. Bella and I. V. Yaschenko, On AP and WAP spaces, Comment. Math. Univ. Carolinae 40 (1999), no. 3, 531-536
  3. A. Dow, M. G. Tkachenko, V. V. Tkachuk and R. G. Wilson, Topologies generated by discrete subspaces, Glasnik Matematicki 37 (2002), no. 57, 187-210
  4. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1970
  5. S. P. Franklin, Spaces in which sequences suffice II, Fund. Math. 61 (1967), 51-56 https://doi.org/10.4064/fm-61-1-51-56
  6. W. C. Hong, A Theorem on countably Frechet-Urysohn spaces, Kyungpook Math. J. 43 (2003), no. 3, 425-431
  7. M. Ismail and P. Nyikos, On spaces in which countably compact sets are closed, and hereditary properties, Topology Appl. 11 (1980), 281-292 https://doi.org/10.1016/0166-8641(80)90027-9
  8. J. Penlant, M. G. Tkachenko, V. V. Tkachuk, and R. G. Wilson, Pseudocompact Why-burn spaces need not be Frechet, Proc. Amer. Math. Soc. 131 (2002), no. 10, 3257-3265 https://doi.org/10.1090/S0002-9939-02-06840-5
  9. L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, Springer-Verlag, Berlin, 1978
  10. V. V. Tkachuk and I. V. Yaschenko, Almost closed sets and topologies they determine, Comment. Math. Univ. Carolinae 42 (2001), no. 2, 395-405
  11. A. Wilansky, Between $T_1 $ and $T_2 $, Amer. Math. Monthly 74 (1967), 261-266 https://doi.org/10.2307/2316017

Cited by

  1. A NOTE ON SPACES DETERMINED BY CLOSURE-LIKE OPERATORS vol.32, pp.3, 2016, https://doi.org/10.7858/eamj.2016.027
  2. A GENERALIZATION OF A SEQUENTIAL SPACE AND RELATED SPACES vol.36, pp.2, 2014, https://doi.org/10.5831/HMJ.2014.36.2.425
  3. ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT vol.25, pp.3, 2010, https://doi.org/10.4134/CKMS.2010.25.3.477