Linearly S-closed spaces

Abstract

In this paper we introduce the notion of linearly S-closed spaces and is shown that it strictly lies between S-closedness and countable S-closedness. The class of linearly S-closed spaces is preserved by irresolute images and regular open as well as regular closed subsets. We characterize linearly S-closed spaces in terms of filters and complete s-accumulation point of families of semi-open sets and study their basic properties. We prove that regular, linearly S-closed spaces are extremally disconnected and hence zero dimensional. Moreover, we prove that every first countable, Hausdorff, linearly S-closed spaces is finite. While regular S-closed spaces are compact there are non compact, regular linearly S-closed spaces. It is established that in class of regular, compact spaces the notions of S-closedness, linearly S-closedness and extremely disconnectedness are equivalent. It is shown that the product of linearly S-closed spaces is not linearly S-closed in general. Several examples are provided to illustrate our results.