- - - Department of Mathematics, College of Education for Pure Science, Ibn AL-Haitham, University of Baghdad, Baghdad-Iraq.
- - - Department of Mathematics, College of Education for Pure Sciences (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq

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.  

Topological spaces, filter base, fibrewise slightly perfect topological spaces