Existence of a homeomorphism from the space of continuous functions to the space of compact Subsets of a topological space, X

This paper presents proof that there exists a subspace of the space of continuous functions on a topological space X, which is homeomorphic to the space of compact subsets of X. Those let C(X) denote the space of continuous functions on a topological space X and K(X) be the space of compact subsets of X. We prove that there exists a subspace of C(K(X)) which is homeomorphic to C(X). The result remains valid for compact open topology and point-wise convergence topology on K(X).