Countably Paracompactness Of Fibrewise Topology

Fibrewise topology is a branch of modern topology theory, which developed very quickly.There are fibrewise versions of many of the important concepts of topology .Among them, the countable paracompactness ,as a paracompactness promotion, holds important status in general topological space,and it has some interesting properties.Also, the fibrewise countable paracompactness ,as a fibrewise paracompactness promotion ,holds important status in fibrewise topological space, accordingly it has some interesting properties. Combined with the analysis of the existed countable paracompactness definition and its correlative properties in general topological space , the aim of the paper is to supplement fibrewise version of the defitition of countable paracompactness and tis correlative properties.In the first part of this paper ,point-fibrewise paracompact which is a weak definition is given by the link between fibrewise topology and general topology ,when the topological base space is limited to one point space.But the manner of the definition can not profoundly reflect the ideas of firewise topology,because the role of the base space is the main difference between fibrewise topology and general topology. In the second part of this paper, the fiberwise countable paracompact space Definition which is not not only taken into account the link between countable paracompact of general topology and each fiber,but also taken into account the neighborhood of the base of the fiberwise space is givened.In the third part of the paper,the author would like to have a systematic way of producting fiberwise locally countable paracompact space,which is not only taken into account the locality of the base space, but also maden fiberwise topological space limit.Meanwhile,Many of the important results we shall be proving in point-fibrewise countable paracompact space, fiberwise countable paracompact space and fiberwise locally countable paracompact space are firewise versions of results which will probably be well knowen to the reader in general topology space.In the last part of this paper ,we discusses that: Besides the coditions which the morphism should satisfy to preserve (inversely preserve) point-fibrewise countable paracompactness,fibrewise countable paracompactness ,fibrewise locally countable paracompactness in the Top* category.(the objects are the fibrewise topological spaces over different base spaces , morphism from X toY is a pair ( f ,λ)).