Recovering a compact Hausdorff space $X$ from the compatibility ordering on $C(X)$
Let $f$ and $g$ be scalar-valued, continuous functions on some topological space. We say that $g$ dominates $f$ in the compatibility ordering if $g$ coincides with $f$ on the support of $f$. We prove that two compact Hausdorff spaces are homeomorphic if and only if there exists a compatibility isomorphism between their families of scalar-valued, continuous functions. We derive the classical theorems of Gelfand–Kolmogorov, Milgram and Kaplansky as easy corollaries to our result, as well as a theorem of Jarosz [Bull. Canad. Math. Soc. 33 (1990)]. Sharp automatic-continuity results for compatibility isomorphisms are also established.