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

The  main  purpose  of  this  paper  is  to  introduce  a  some  concepts  in  fibrewise  totally topological space which are called fibrewise totally mapping, fiberwise totally closed mapping, fibrewise weakly totally closed mapping, fibrewise totally perfect   mapping fibrewise almost totally perfect  mapping.  Also  the  concepts as  totally adherent  point, filter, filter  base, totally converges to a subset, totally directed toward a set, totally rigid, totally-H-set, totally Urysohn space, locally totally-QHC totally topological space are introduced and the main concept in this paper  is  fibrewise  totally  perfect  mapping  in  totally  topological  space.  Several  theorem  and characterizations concerning with these concepts are studied.

Fibrewise totally topological space, filter base, totally converges, totally closed mapping, totally rigid a set, totally perfect mapping