Rercovering a compact Hausdorff space X from the compatibility ordering on C(X)

Abstract

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.

Keywords

• compatibility ordering
• compatibility isomorphism
• Gelfand–Kolmogorov theorem
• Banach–Stone theorem
• Milgram’s theorem
• Kaplansky’s theorem
• lattice
• ultrafilter in a lattice
• compact Hausdorff space
• automatic continuity