GENERALIZED FRÉCHET-URYSOHN SPACES

Title & Authors
GENERALIZED FRÉCHET-URYSOHN SPACES
Hong, Woo-Chorl;

Abstract
In this paper, we introduce some new properties of a topological space which are respectively generalizations of $\small{Fr\{e}chet}$-Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, $\small{Fr\{e}chet-Urysohn}$, first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably $\small{Fr\{e}chet-Urysohn}$. We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is $\small{Fr\{e}chet-Urysohn}$. We finally obtain a sufficient condition for the ACP closure operator $\small{[{\cdot}]_{ACP}}$ to be a Kuratowski topological closure operator and related results.
Keywords
$\small{Fr\{e}chet-Urysohn}$;sequential;countably $\small{Fr\{e}chet-Urysohn}$;countable tightness;AP;countably AP;WAP;ACP;WACP;countably compact;
Language
English
Cited by
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References
1.
A. V. Arhangel'skii, The frequency spectrum of a topological space and the product operation, Trans. Moscow Math. Soc. 2 (1981), 163-200

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.
J. Dugundji, Topology, Reprinting of the 1966 original. Allyn and Bacon Series in Advanced Mathematics. Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978

6.
S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115

7.
G. Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, 423-501, North-Holland, Amsterdam, 1984

8.
W. C. Hong, A note on weakly first countable spaces, Commun. Korean Math. Soc. 17 (2002), no. 3, 531-534

9.
W. C. Hong, A theorem on countably Frechet-Urysohn spaces, Kyungpook Math. J. 43 (2003), no. 3, 425-431

10.
E. A. Michael, A quintuple quotient quest, General Topology and Appl. 2 (1972), 91-138

11.
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

12.
T. W. Rishel, A class of spaces determined by sequences with their cluster points, Portugal. Math. 31 (1972), 187-192

13.
F. Siwiec, On defining a space by a weak base, Pacific J. Math. 52 (1974), 233-245

14.
L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, Second edition. Springer-Verlag, New York-Heidelberg, 1978

15.
V. V. Tkachuk and I. V. Yaschenko, Almost closed sets and topologies they determine, Comment. Math. Univ. Carolin. 42 (2001), no. 2, 395-405

16.
J. E. Vaughan, Countably compact and sequentially compact spaces, Handbook of settheoretic topology, 569-602, North-Holland, Amsterdam, 1984