Paracompactness Of Fibrewise Topology

The fibrewise viewpoint is extensively adopted in the general topological theory. In fact, there appear to be fibrewise versions of many of the major concepts of topology . I.M.James has given the definitions of fibrewise compact and fibrewise locally compact. The aim of this paper is to supplement fibrewise version of the definition of paracompact and its properties which are familiar to us in the general topology. Therefore , the approach can be regarded as a Fibrewise version of the paracompact in the general topology or generalization of fibrewise compact.In the first section of this paper, point fibrewise paracompact which is weak definition is provided by the link between fibrewise topology and general topology. Meanwhile, this article discuss some important properties, such as hereditary with respect to closed subsets, the coproduct of point fibrewise paracompact spaces is point fibrewise paracompact, the product of point fibrewise paracompact space and fibrewise compact space is point fibrewise paracompact, every point fibrewise paracompact Hausdorff space is fibrewise regular and normal, invariance of point fibrewise paracompact under closed mapping, inverse invariant of point fibrewise paracompact under proper mapping.However, the approach of the definition can not profoundly reflect the ideas of fibrewise topology. The main difference between fibrewise topology and general topology is the role of base space, such as the method of defining fibrewise compact and fibrewise local compact by I.M.James. Therefore, in the second section of this paper, the author would like to have a systematic way of producting fibrewise paracompact spaces. At the same time, its properties also are discussed in this paper, such as hereditary with respect to closed subsets, the coproduct of fibrewise paracompact spaces is fibrewise paracompact, the product of fibrewise paracompact space and fibrewise compact space is fibrewise paracompact, every fibrewise paracompact Hausdorff space is fibrewise regular and normal, invariance of fibrewise paracompact under closed mapping, inverse invariant of fibrewise paracompact under proper mapping.The last part of this article discusses invariance and inverse invariant of fibrewise paracompact in the TOP? category(the objects are the fibrewise topological spaces over different base spaces, the mophism from X toY is a pair ( f ,λ)). The following are major conclusions:(1) mophism ( f ,λ),point fibrewise paracompact Hausdorff and fibrewise paracompact Hausdorff is invariant when f is closed fibrewise function andλis open mapping.(2) mophism ( f ,λ),point fibrewise paracompact and fibrewise paracompact is inverse invariant when f is proper fibrewise function andλis injective.