This paper aims to define fuzzy pretopogenous structure based on way below
relation (or an L-fuzzifying pretopogenous structure (LFPT structure, for
short)) and study some of its properties. Also, the concepts of L-fuzzifying
pre-neighborhood, L-fuzzifying pre-interior, and L-fuzzifying pre-closure
operators are established and we used these concepts to build an
L-fuzzifying topology. Furthermore, a natural link is established between
L-fuzzifying pretopogenous and L-fuzzifying topology. Finally, the maps
between L-fuzzifying pretopogenous structures and initial fuzzifying
structures are investigated.