Order isomorphisms on function spaces (Q2866761)

Let \(X\) be a compact Hausdorff space. A subspace \(A\) of continuous functions is said to precisely separate points from closed sets if, given a closed set \(F \subset X\) and \(x \notin F\), there exists an \(f \in A\), \(0\leq f \leq 1\), \(f = 0\) on \(F\) and \(f(x) = 1\). For compact spaces \(X,Y\) and subspaces \(A \subset C(X)\) and \(B \subset C(Y)\), containing constants and precisely separating points from closed sets, the authors show that any order preserving linear onto isomorphism \(T: A \rightarrow B\) is of the form \(T(f) = T(1)f \circ h^{-1}\) for a surjective homeomorphism \(h: X \rightarrow Y\). The proof follows a standard Banach-Stone theorem-type argument (see [\textit{E. Behrends}, M-structure and the Banach-Stone theorem. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0436.46013)]) by analysing the zero sets. NEWLINENEWLINENEWLINEThe paper also contains an analysis of these aspects for metric and completely regular topological spaces.