More on strongly compact spaces
by
Saeid Jafari
Department of Mathematics and Physics, Roskilde University, Postbox 260, 4000 Roskilde, Denmark
Coauthors: Takashi Noiri (Yatsushiro College of Technology)

It is well-known that the effect of the investigation of properties of closed bounded intervals of real numbers, spaces of continuous functions and solutions to differential equations are the possible motivations for the formation of the notion of compactness which is now one of the most important, useful and fundamental notions of not only general topology but also the other advance branches of mathematics. Many researchers have pithly studied the fundamental properties of compactness which now the results can be found in any undergraduate textbook on analysis and general topology. The productivity and fruitfulness of the notion of compactness motivated many researchers to generalize this notion. In the course of these attempts, many stronger and weaker forms of compactness introduced and investigated. In 1982, Atia et al. (1) introduced a strong version of compactness defined in terms of preopen subsets of a topological space. In 1984, Mashhour et al. (4) introduced the notion of strongly compact relative to a topological space and established several characterizations of these spaces. In 1987, Ganster (2) obtained an interesting result that there exist no infinite spaces which are both strongly compact and semi compact. He also proved that a topological space is strongly compact if and only if it is compact and that every infinite subset of X has nonempty interior. In 1988, Jankovic et al. (3) have shown that a topological space (X, T) is strongly compact if and only if it is compact and the family of dense sets in (X, T) is finite. The object of this talk is to give some characterizations of strongly compact spaces in terms of nets and filterbases. We also introduce the notion of pre-complete accumulation point by which we give some characterizations of strongly compact spaces. By introducing the notions of 1-lower (resp. 1-upper) precontinuous functions and the known notions of 1-lower(resp. 1-upper) compatible partial orders we investigate some more properties of strong compactness.

(1)R. H. Atia, S. N. El-Deeb and I. A. Hasanein, A note on strong compactness and S-closedness, Mat. Vesnik 6(19)(1982), 23-28.
(2) M. Ganster, Some remarks on strongly compact spaces and semi compact spaces, Bull. Malaysia Math. Soc. (10)2(1987), 67-81.
(3) D. S. Jankovic, I. Reilly and M. K. Vamanamurthy, On strongly compact topological spaces, Q A in General Topology, vol. 6(1988), 29-39.
(4) A. S.Mashhour, M. Abd El-Monsef, I. A. Hasanein and T. Noiri, Strongly compact spaces, Delta J. Sci. 8(1984), 30-46.

Date received: April 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-07.