On the continuity of lattice isomorphisms on C(X, I)

Abstract

We investigate topological conditions on a compact Hausdorff space Y, such that any lattice isomorphism φ: C(X, I) -> C(Y, I), where X is a compact Hausdorff space and I is the unit interval [0, 1], is continuous. It is shown that in either of cases that the set of G_δ points of Y has a dense pseudocompact subset or Y does not contain the Stone-Čech compactification of N, such a lattice isomorphism is a homeomorphism.