The Countability Of Fiber Hyperspace

The countability is one of basic topological properties. It holds the certainly important status in the hyperspace theory research. The main purpose of this paper is supplementing the definition of countability in the fiber hyperspace theory. Combined with the definitions of fibrewise first countability and fibrewise second countability, we get the definition of fibrewise compact set first countability.We get the conclution that if C(X) is fibrewise first countability, X is fibrewise compact set first countability. C(X) is fibrewise second countability iff X is, fibrewise second countability. Combined fibrewise separation conditions, it obtain some relations of fibrewise first countability, fibrewise second countability and fibrewise separation conditions on fibrewise hyperspace.On the basis of the definition of fibrewise normal we get the definition of fibrewise weak normal.We proved some conclutions such as follows:X is fibrewise hausdorff iff C(X) is fibrewise hausdorff. X is fibrewise regular iff C(X)is fibrewise regular. If C(X) is fibrewise second countability and fibrewise normal,X is fibrewise weak normal.The paper discussed the relations of fibrewise countability, fibrewise Lindel of property, compactness, Frechet space and sequential space on fibrewise hyperspace. It proved that if C(X)is fibrewise first countability on C(B) and C(B) is first countability, X is a Frechet space.