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

In this paper we define and study new concepts of fibrwise totally topological spaces over B namely fibrewise totally compact and fibrwise locally totally compact spaces, which are generalization of well known concepts totally compact and locally totally compact topological spaces. Moreover, we study relationships between fibrewise totally compact (resp, fibrwise locally totally compact) spaces and some fibrewise totally separation axioms.

Fiberwise totally topological spaces, Fiberwise totally compact spaces, fiberwise locally totally compact spaces, fibrewise totally separation axioms