Countably s-closed spaces and feebly s-compact spaces
by
G. B. Navalagi
Department of Mathematics, G.H. College, Haveri-581110, Karnataka, India

In 1987, G.D. Maio and T. Noiri [Indian J Pure Appl. Math. 18(3), March (1987), 226-233]] have defined and studied the s-closed spaces which is generalization of S-closed space due to T. Thampson [PAMS, 60(1976), 335-338]. In this paper, using semiopen sets of N. Levine [AMM, 70(1963), 36-41] and semiregular sets due to G.D. Maio et [cf . above ref.], a set A is called semiregular if it is both semiopen and semiclosed: We define and study the concepts like countably s-closed spaces, a space X is called coutably s-closed if every countable semiopen cover of X has a finite subfamily the semiclosures of whose members cover X and feebly s-compact spaces, a space X is called feebly s-compact if every countable open cover of X has a finite subfamily the semiclosure of whose members cover X. Among other things it is proved that:

1. Every s-closed space is countably s-closed,
2. Every countably s-closed space is feebly s-compact space,
3. A space X is called countably s-closed if every countable cover of X by semiregular sets of X has a finite subcover and
4. Countably s-closedness is semiregular hereditary.

Also, we study invariant theorems of these spaces and comparison with the S-closed spaces in topology.

Date received: June 24, 2000