Locally Closed Sets, Submaximal Spaces and Some Other Related Concepts

A subset A of a topological space X is called locally closed if A = G ∩ B, where G is an open subset and B is a closed subset of X; X is called submaximal if every subset of X is locally closed. In this paper, we show that if βX, the Stone-Cech compactification of ˇ X, is a submaximal space, then X is a compact space and hence βX = X. We observe that every submaximal Hausdorff space is an ncd-space (a space in which does not have a nonempty compact and dense in itself subset). It turns out that every dense in itself Hausdorff space is pseudo-finite if and only if it is a (cei, f)-space (a space in which every compact subspace of X with empty interior is finite). A new characterization for submaximal spaces is given. Given a topological space (X, T ), the collection of all locally closed subsets of X forms a base for a topology on X which is denoted by Tl. We study some relations between (X, T ) and (X, Tl). For example, we show that (X, T ) is a locally indiscrete space if and only if T = Tl.