© Hindawi Publishing Corp. COUNTABLY I-COMPACT SPACES

Abstract. We introduce the class of countably I-compact spaces as a proper subclass of countably S-closed spaces. A topological space (X,T) is called countably I-compact if every countable cover of X by regular closed subsets contains a finite subfamily whose interiors cover X. It is shown that a space is countably I-compact if and only if it is ex-tremally disconnected and countably S-closed. Other characterizations are given in terms of covers by semiopen subsets and other types of subsets. We also show that countable I-compactness is invariant under almost open semi-continuous surjections. 2000 Mathematics Subject Classification. 54D20, 54G05. 1. Introduction. A topological space (X,T) is called S-closed by Thompson [8] if every cover ofX by semiopen subsets contains a finite subfamily whose union is dense in (X,T). Cameron [1] showed that (X,T) is S-closed if and only if every cover of X by regular closed sets has a finite subcover (accordingly, S-closed spaces are called rc-compact). A topological space (X,T) is called I-compact by Cameron [2] if ever