The Fiberwise Compactness Of Fiberwise Hyperspace Topological Space

Fibrewise topology ,as one of the fast branches in the modem topology theory,is favored by the growing number of scholars.At the same time, Hyperspace, as a special kind of topological space, also has very important research value. Fibrewise topology space take a topological space as its base, so this paper will give a new definition of topological spaces called fibrewise hyperspace topological space ,which was produced by the concept of hyperspace and fibrewise topology space .Meanwhile,the basis space and the fibrewise space is its corresponding hyperspace respectively, then this paper will discuss the compactness of fibrewise hyperspace topological space.This paper mainly discuss the following aspects:First, combine with the way of the definition of fiberwise topological space, Both base space B and fiberwise space X are replaced respectively with the hyperspace which consisting of all the corresponding non-empty compact sets or non-empty closed sets. Under the both situations, By studying the projection whether continuous, get two forms of definitions of fiberwise hyperspace topological space.Second, the compactness, as an important property of fiberwise topological space ,there are many important research results, this paper will combine with the existing properties of compact in fiberwise space, By corresponding to discuss the compact in fiberwise hyperspace topological space.Third, many fiberwise topological properties are discussed in the category of ,By analogying the way of definition of TOP_B category, get the definition of category of in fiberwise hyperspace topological space. Combined with an example to discuss the relationships of the compact of the two fiberwise hyperspace topological spaces in the TOP_C(B) category .The main conclusions of this paper are as follows:Theorem2.2.2: If (C_B(X), T_γ) is fiberwise compact, B is compact,then (X_B,T) is fiberwise compact.Theorem2.2.3: If the map p : X→B is proper, X is regular,then (C_B(X), T_γ)is fiberwise compact.Theorem 4.2.1: If (2_B^X ,T_γ) is fiberwise compact, then (X_B,T) is fiberwise compact.