\(C(X)\) related to its topological homeomorphism group (Q960849)

For a topological space \(X\), let \(C(X)\) denote the space of continuous real-valued functions on \(X\) with the uniform topology, and let \(\mathcal{H}(X)\) denote the space of homeomorphisms on \(C(X)\) with the fine topology. (It is known that \(\mathcal{H}(X)\) is in fact a topological group.) The author considers the following problem: For spaces \(X\) and \(Y\) and given subgroups \(\mathcal{F}\) and \(\mathcal{G}\) of \(\mathcal{H}(C(X))\) and \(\mathcal{H}(C(Y))\), respectively, to find homeomorphisms between \(C(X)\) and \(C(Y)\) that induce topological isomorphisms between \(\mathcal{F}\) and \(\mathcal{G}\), and conversely, to find topological isomorphisms between \(\mathcal{F}\) and \(\mathcal{G}\) that induce homeomorphisms between \(C(X)\) and \(C(Y)\). In the so-called Correspondence Theorem, the author shows that for certain \(\mathcal{F}\) and \(\mathcal{G}\) there is a natural one-to-one correspondence between a certain set of topological isomorphisms from \(\mathcal{F}\) onto \(\mathcal{G}\) and a certain set of homeomorphisms from \(C(X)\) onto \(C(Y)\) that relate to \(\mathcal{F}\) and \(\mathcal{G}\). Also the Weak Correspondence Theorem is given that is satisfied by a larger family of subgroups \(\mathcal{F}\) and \(\mathcal{G}\), but whose correspondence is only an injection rather than a bijection. A number of examples are presented of types of subgroups of \(\mathcal{F}\) and \(\mathcal{G}\) that satisfy the assumptions of either the Correspondence Theorem or the Weak Correspondence Theorem.