The Hyperspaces Of Continuous Functions

It is well known that the classical method to study a space of continous functions is they are endowed with the uniformly converging topology, compact-open topology or the topology of pointwise convergence. The basic thought of this paper is that when a continuous function is regarded as a closed subset of the corresponding product space, the family of continuous functions becomes a subspace of the hyperspace of closed subsets based on the product space. The topology on the hyperspace here is the known Vietoris topology. So the space of continuous functions is naturally said to be the hyperspace of continuous functions.This paper consists of two chapters.In the first chapter, we first state briefly the developing history of infinite-dimensional topology and then generalize the background of the study on the spaces of continuous functions, in which a series of results about the spaces of continuous functions when they are endowed with various natural topologies have been contained.The results about the hyperspaces are introduced in the second chapter. Based on the thought that the family of all the continuous functions from a infinite locally connected compact metrizable space X to a infinite locally path-connected compact metrizable space Y is regarded as a subspace of the hyperspace Cld(X × Y) of closed subsets in the product space X×Y, when X is the Hilbert cube Q, we discuss the topological structures of C(X, Q) and its clo-sure C(X,Q) in Cld(X × Q), moreover, we have (C(X,Q),C(X,Q)) ≈ (Q,s); and when X is the unit interval I, we discuss the closure C(I, Y) of C(I, Y) in Cld(X × Y), whose elements achieves topological characterization. Lastly, we illustrate the main reasons here X is confined to I.