Remarks on Almost Locally Connected Spaces

Abstract

Abstract. Some preservation theorems for almost local connectedness are proved. IntroductionIn 1981, as a generalization of locally connected topological spaces, the concept of almost locally connected spaces was introduced and studied by Mancuso [5]. Subsequently, Noiri [8] extended the work of Mancuso. In 1985, Jankovic [3] generalized and improved Noiri's results and among others, obtained the following result: Let f : X -• Y be a weakly continuous, almost open surjection. If X is almost locally connected, then so is Y.Mrsevic, Reilly and Vamanamurthy [6] and Saleemi, Shahzad and Alghamdi [9] studied this notion and made their contribution to the subject. The aim of this paper is to prove a general preservation theorem for almost locall connectedness and to generalize the above result of Jankovic [3]. PreliminariesThroughout this paper spaces mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X. The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. The set A is called regular open (resp. regular closed) if Int(Cl(A)) = A (resp. Cl(Int{A)) = A). The family of all regular open subsets of a space X is represented by RO(X). The family RO(X) is a basis for a topology on X and the set X with this topology is called the semi-regularization of X and is denoted by X s . A space X is called almost locally connected at a; € A" if for each G G RO(X) containing x there exists an open connected set V such that x G V C G. X is almost locally connected if it is almost locally connected at each of its points. Every locally connected space is almost locally connected; however the converse is not true Note that continuity => almost continuity =>• closure continuity => weak continuity and none of these implications is reversible. For any AC X, we define : for each open set U in X. Clearly every almost open surjection is s-almost open. But the converse is false [9]. Main resultsWe shall need the following results in the sequel. LEMMA 3.1 ([11]). Let f : X -> Y be a connected function and C be a component ofY. Then / -1 (C) is a union of some connected components of X. LEMMA 3.2 ([8]). A space X is almost locally connected if and only if the components of regular open sets in X are regular open.Now we prove our main results. If X is almost locally connected, then so isY. Proof. Let V € RO(Y) and C a component of V. Then by Lemma 3.1, f~l(C) is a union of some components of / -1 (F). As in [11], / -1 (C) fl Int{Cl(f~l(V)))is a union of some components of Int{Cl{f~l{V))). Since X is almost locally connected, it follows that f~l(C) fl Int{Cl(f~l{V))) is UyeC, then / _1 (y) C / _1 (C). Since C C V, by condition (S) we have that f-\y) Q Int{f~l{V)).Thus /-%) C f-\C)f\Int{Cl{r\V))) and so C C f*(f-\C)nlnt(Cl(f-\V)))).On the other hand,