ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT

Title & Authors
ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT
Hong, Woo-Chorl;

Abstract
In this paper, we introduce a new property (*) of a topological space and prove that if X satisfies one of the following conditions (1) and (2), then compactness, countable compactness and sequential compactness are equivalent in X; (1) Each countably compact subspace of X with (*) is a sequential or AP space. (2) X is a sequential or AP space with (*).
Keywords
Fr$\small{\acute{e}}$chet-Urysohn;sequential;AP;WAP;countable tightness;weakly discretely generated;compact;countably compact;sequentially compact and property(*);
Language
English
Cited by
References
1.
A. V. Arhangel'skii, Topological Function Spaces, Mathematics and its Applications (Soviet Series), 78. Kluwer Academic Publishers Group, Dordrecht, 1992.

2.
A. V. Arhangel’skii and L. S. Pontryagin (Eds.), General Topology I, Encyclopaedia of Mathematical Sciences, vol.17, Springer-Verlage, Berlin, 1990.

3.
A. Bella, On spaces with the property of weak approximation by points, Comment. Math. Univ. Carolin. 35 (1994), no. 2, 357–360.

4.
A. Bella and I. V. Yaschenko, On AP and WAP spaces, Comment. Math. Univ. Carolin. 40 (1999), no. 3, 531–536.

5.
A. Dow, M. G. Tkachenko, V. V. Tkachuk, and R. G. Wilson, Topologies generated by discrete subspaces, Glas. Mat. Ser. III 37(57) (2002), no. 1, 187–210.

6.
J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1970.

7.
S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107–115.

8.
S. P. Franklin, Spaces in which sequences suffice. II, Fund. Math. 61 (1967), 51–56.

9.
G. Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, 423–501, North-Holland, Amsterdam, 1984.

10.
W. C. Hong, Generalized Fr´echet-Urysohn spaces, J. Korean Math. Soc. 44 (2007), no. 2, 261–273.

11.
W. C. Hong, On spaces in which compact-like sets are closed, and related spaces, Commun. Korean Math. Soc. 22 (2007), no. 2, 297–303.

12.
J.-I. Nagata, Modern General Topology, North-Holland Publishing Co., Amsterdam-London, 1974.

13.
F. Obersnel, Some notes on weakly Whyburn spaces, Topology Appl. 128 (2003), no. 2-3, 257–262.

14.
J. Pelant, M. G. Tkachenko, V. V. Tkachuk, and R. G. Wilson, Pseudocompact Whyburn spaces need not be Frechet, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3257–3265.

15.
L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, Springer-Verlag, New York-Heidelberg, 1978.

16.
V. V. Tkachuk and I. V. Yaschenko, Almost closed sets and topologies they determine, Comment. Math. Univ. Carolin. 42 (2001), no. 2, 395–405.

17.
J. E. Vaughan, Countably compact and sequentially compact spaces, Handbook of settheoretic topology, 569–602, North-Holland, Amsterdam, 1984.

18.
A. Wilansky, Topology for Analysis, Waltham, Mass.-Toronto, Ont.-London, 1970.