Mappings and decompositions of continuity on almost Lindelöf spaces

Among the various covering properties of topological spaces a lot of attention has been given to those covers which involve open and regularly open sets. In 1982 Balasubramanian [4] introduced and studied the notion of nearly Lindelöf spaces and in 1984 Willard and Dissanayake [21] gave the notion of almost Lindelöf spaces. Then in 1996 Cammaroto and Santoro [5] studied and gave further new results about these spaces which are considered as one of the main generalizations of Lindelöf spaces. Moreover, decompositions of continuity have been recently of major interest among general topologists. They are being studied by many authors, including Singal and Singal [19], Popa and Stan [16], Noiri [14], Long and Herrington [11], Mashhour et al. [12], Abd El-Monsef et al. [1], Dontchev [6], Dontchev and Przemski [7], Nasef and Noiri [13], Park and Ha [15], and Baker [2, 3]. In fact, mathematicians introduced in several papers different and interesting new decompositions of continuity as well as generalized continuous functions. Throughout the present paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated. The topological space (X ,τ) will be replaced by X if there is no chance for confusion. The interior and the closure of any subset A of (X ,τ) will be denoted by Int(A) and Cl(A), respectively. The purpose of this paper is to study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces. We also show that some mappings preserve this property. The main result is that the image of an almost Lindelöf space under a θ-continuous function is almost Lindelöf.