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

Title & Authors
ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES
Hong, Woo-Chorl;

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 $\small{Fr\acute{e}chet}$-Urysohn with unique sequential limits, for weakly discretely generated AP-spaces, unique sequential limits $\small{{\equiv}KC{\equiv}C-closed{\equiv}SC-closed}$, 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
KC;C-closed;SC-closed;$\small{Fr\acute{e}chet}$-Urysohn;sequential;AP;WAP;weakly discretely generated;
Language
English
Cited by
1.
ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT,;

대한수학회논문집, 2010. vol.25. 3, pp.477-484
2.
A GENERALIZATION OF A SEQUENTIAL SPACE AND RELATED SPACES,;;

호남수학학술지, 2014. vol.36. 2, pp.425-434
1.
A NOTE ON SPACES DETERMINED BY CLOSURE-LIKE OPERATORS, East Asian mathematical journal, 2016, 32, 3, 365
2.
A GENERALIZATION OF A SEQUENTIAL SPACE AND RELATED SPACES, Honam Mathematical Journal, 2014, 36, 2, 425
3.
ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT, Communications of the Korean Mathematical Society, 2010, 25, 3, 477
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

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

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

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