related to its topological homeomorphism group

When the space C ( X ) of continuous real-valued functions on X has the uniform topology, the space H ( C ( X ) ) of homeomorphisms on C ( X ) is a topological group when it has the fine topology. This article shows that for certain subgroups F and G of H ( C ( X ) ) and H ( C ( Y ) ) , respectively, there is a natural one-to-one correspondence between a certain set of topological isomorphisms from F onto G and a certain set of homeomorphisms from C ( X ) onto C ( Y ) that relate to F and G . A number of examples are given of types of subgroups of H ( C ( X ) ) that satisfy this Correspondence Theorem and a weaker version of it.