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

- Journal title : Communications of the Korean Mathematical Society
- Volume 25, Issue 3, 2010, pp.477-484
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2010.25.3.477

Title & Authors

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

Hong, Woo-Chorl;

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

Frchet-Urysohn;sequential;AP;WAP;countable tightness;weakly discretely generated;compact;countably compact;sequentially compact and property(*);

Language

English

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.

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.

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.