| The fibrewise separation condition holds important status in the TOPB Category ( the objects are the fibrewise topological spaces over B , for two objects ( X, p) and (Y , q), a morphism from X to Y is a continuous map such that p = qoΦ). It has some interesting properties. Namely,in the TOPB category, a morphism preserves ( inversely preserves ) the fibrewise separation condition when it satisfies certain condition. Combined with the analysis of the existed properties, the conditions of the morphism which also preserves ( inversely preserves ) the fibrewise separation condition in the TOP2B category ( the object are the fibrewise topological spaces over different bases, for two objcets a morphism from 2X to 2Y is a pairΦ,λof continuous maps such thatλo f= goΦ) are given .Here is the outline ofthesis:1,morphism(Φ,λ), fibrewise R0,fibrewise Ti (i =0,1,2), fibrewise (completely) regular,fibrewise normal are preserved whenΦis open continuous fibrewise surjection andλis open continuous map.2,morphism(Φ,λ),fibrewise R 0 and fibrewise (eompleteyl) regular are inversely preserved whenΦis open continuous fibrewise surjection andλis continuous mapping; fibrewise normal is inversely preserves whenΦis closed continuous fibrewise injection andλis continuous map.3, What conditions should be hold when the fiberwise over a subspace of a space can be hereditary. |
