Local Isomorphisms of Algebras of Continuous Functions

This paper is concerned with the problem of when a local isomorphism of C ( X ) and C ( Y ) implies a local homeomorphism of X and Y . More precisely, if X and Y are locally compact Hausdorff spaces, if p ɛ Y and q ɛ Y , and if C ( X ) P is isomorphic to C ( Y ) q , does p have a neighbourhood which is homeomorphic to a neighbourhood of q ? The answer is negative in general, but is shown to be positive in two cases: when X and Y are metric and the isomorphism of C ( X ) P and C ( Y ) q arises from a continuous map of Y into X , and when X and Y are manifolds. The proof for manifolds proceeds by showing that the local topological dimension of X at p is an invariant of the ring C ( X ) P , which is proved with the aid of the lattice of germs of closed sets at p .