The aim of this paper is to introduce fuzzy ΛrgΧ -set and to study some of its generalized forms. The structure constituted by fuzzy ΛrgΧ -set is also included in the treatise. Moreover we define generalized fuzzy gx -closed set and fuzzy ΛrgΧ-closed set, which are the weaker forms of fuzzy gx-closed set. The above two weaker forms are independent to each other but jointly they imply fuzzy gx -closed set. Finally, in addition, as the application of fuzzy ΛrgΧ -closed set, we study fuzzy (ΛrgΧ , gY)- continuity and fuzzy (ΛrgΧ gf - gX , gY)
-continuity and a few of their properties. In general, fuzzy contra ( gX , gY -continuity does not imply fuzzy ( gX , gY ) -continuity, but this paper provides one condition under which the fact is true.
