Integral extensions on rings of continuous functions

Abstract

Let π : X → Y be a surjective continuous map between Tychonoff spaces. The map π induces, by composition, an injective morphism C ( Y ) → C ( X ) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C ( Y ) as a subring of C ( X ) . This paper deals with finiteness properties of the ring extension C ( Y ) ⊆ C ( X ) in relation to topological properties of the map π : X → Y . The main result says that, for X a compact subset of R n , the extension C ( Y ) ⊆ C ( X ) is integral if and only if X decomposes into a finite union of closed subsets such that π is injective on each one of them.