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.