This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy
topologies), introduced by Ying [1]. It investigates topological notions defined by means of a-open sets
when these are planted into the framework of Ying's fuzzifying topological spaces (by Łukasiewicz
logic in [0, 1]) . The concept of a-irresolute functions and a-compactness in the framework of fuzzifying topology are introduced and some of their properties are obtained. Weuse the finite intersection property to give a characterization of fuzzifying a-compact spaces. Furthermore, we study the image of fuzzifying a-compact spaces under fuzzifying a-continuity and fuzzifying a-irresolute maps.