Biseparating Maps and Homeomorphic Real-Compactifications

If X and Y are realcompact and T: C(X) → C(Y) is a ring isomorphism then two things happen: (1) there is a homeomorphism h of X onto Y and (2) T is a composition map: for all ƒ in C(X), Tƒ = ƒ ∘ h. We strengthen that result in Propositions 2 and 3. We discard the multiplicative property of T and assume only that T is biseparating in the sense that ƒg = 0 ⇔ TƒTg = 0. It remains true that X is homeomorphic to Y but this time, if T is also linear, it is a weighted composition map: for some weight function a in C(Y), Tƒ = a(ƒ ∘ h).