On some classes of sets in extremally disconnected spaces

Abstract

Abstract. In the present paper several characterizations of the classical notion of extremally disconnected spaces are obtained. A few relationships for finite products of extremally disconnected spaces are also studied. IntroductionExtremally disconnected topological spaces were introduced by Gillman and Jerison [6]. It is well-known that each extremally disconnected (e.d., in brief) compact and Hausdorff space is called a Stonean space (for instance, the Stone-Čech compactification βN is a compact, Hausdorff and e.d. space). E.d. spaces play an important role in the theory of Boolean algebras and in some branches of functional analysis. There is a duality between Stonean spaces and the category of complete Boolean algebras. The importance of e.d. space becomes clearer in a study of absolute topological spaces. The purpose of this paper is to give several characterizations of e. [8,9,16,18]), where no separation axioms are assumed. Nevertheless, it is worth to observe that some finite spaces that we make use of as examples, may fulfil weaker forms of separation. For example: the space in Example 1 is semi-T 2 [10] (but not even T 1 ). PreliminariesA topological space on which no separation axiom is assumed will be denoted by (X, τ ). For a subset S of (X, τ ) the closure of S and the interior of S (in (X, τ )) are denoted by cl (S) and int (S) respectively. A subset S of The family of all α-open [resp. preopen; semiopen; semi-preopen] subsets of (X, τ ) is denoted by α(X, τ ) (or τ α ) [resp. PO (X, τ ); SO (X, τ ); SPO (X, τ )]. The family α(X, τ ) is a topology on X larger than τ , in general [12]. The complement of an α-open [resp. semiopen] subset of (X, τ ) is called α-closed [resp. semi-closed] in (X, τ ). The family of all closed [resp. semi-closed] subsets of (X, τ ) will be denoted by C (X, τ ) [resp. SC (X, τ )]. The intersection of all semi-closed subsets of (X, τ ) containing a certain set S ⊂ X is called semi-closure of S and is denoted by scl (S). Analogously, the semi-interior of S ⊂ X, denoted by sint (S), is the union of all semi-open subsets of (X, τ ) contained in S.The following equalities are also known [3, Theorem 1.6]:. The family of all regular open [resp. regular closed] subsets of (X, τ ) is denoted by RO (X, τ ) [resp. RC (X, τ )]. By definition we put R (X, τ ) = RO (X, τ ) ∩ RC (X, τ ). The class R (X, τ ) is the class of all clopen subsets of (X, τ ), that is R (X, τ ) = C (X, τ ) ∩ τ . The following known results will be useful in the sequel. By [1, Theorem 1.5(a), (b)] we have scl (S) = S ∪ int (cl (S)) and (resp.) sint (S) = S ∩ cl (int (S)). This theorem implies at once that scl (S) = int (cl (S)) iff S ∈ PO (X, τ ) [8, Proposition 2.7(a)]. A space (X, τ ) is extremally disconnected if cl (G) ∈ τ for each G ∈ τ . D. S. Janković proved, [7, Theorem 2.9], that (X, τ ) is e.d. iff SO (X, τ ) ⊂ τ α . By ∆ we designate the symmetric difference of two sets. Let us remark that by using of [17, Lemma 1], we can obtain another proof of this result. Before restating Theorem 1 in some ot...