The primary objective of this paper is to present a new concept of fibrewise topological spaces over B
is said to be fibrewise slightly topological spaces over B. Also, we introduce the concepts of fibrewise
slightly perfect topological spaces, filter base, contact point, slightly convergent, slightly directed
toward a set, slightly adherent point, slightly rigid, fibrewise slightly weakly closed, H.set, fibrewise
almost slightly perfect, slightly∗
.continuous fibrewise slightly∗
topological spaces respectively, slightly
Te, locally QHC, In addition, we state and prove several propositions related to these concepts.