On a finer topological space than τ θ and some maps

In 1943, Fomin 7 introduced the notion of ?-continuity. In 1966, the notions of ?-open subsets, ?-closed subsets and ?-closure were introduced by Veli?ko 18 for the purpose of studying the important class of H-closed spaces in terms of arbitrary filterbases. He also showed that the collection of ?-open sets in a topological space (X; ?) forms a topology on X denoted by ? ? (see also 12). Dickman and Porter 4, 5, Joseph 11 continued the work of Veli?ko. Noiri and Jafari 15, Caldas et al. 1 and 2, Steiner 16 and Cao et al 3 have also obtained several new and interesting results related to these sets. In this paper, we will off a finer topology on X than ?? by utilizing the new notions of ??-open and ??-closed sets. We will also discuss some of the fundamental properties of such sets and some related maps.