Factorization and extension of isomorphisms on to homeomorphisms on hyperspaces

For a binormal space X, the space C ( X ) of continuous real-valued functions on X under the graph topology is a topological group that is a subspace of a certain hyperspace L ( X ) under the Vietoris topology. Conditions are given showing when a positive topological group isomorphism from C ( X ) onto C ( Y ) has an extension to an ordered homeomorphism from L ( X ) onto L ( Y ) . It is then shown how such an ordered homeomorphism can be uniquely factored through a homeomorphism from X onto Y. Finally, the existence of such an extension and the existence of such a factorization (restricted to C ( X ) ) are shown to be equivalent.