ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

Single generated algebras of continuos functions

J. M. Domínguez and M. A. Mulero Universidad de Extremadura, Spain

Every continuous map X → Y defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(Y) → C(X). By means of this homomorphism, C(X) is a C(Y)-algebra.

In this work, we study when is C(X) single generated as C(Y)-algebra, that is, when does exist a function f ∈ C(X) such that C(X) = C(Y)[f].

We shall prove, for compact spaces X and Y, that if C(X) = C(Y)[f], then the map X → Y is locally injective.

We shall give examples of locally injective continuous maps, between compact spaces X → Y such that C(X), with the structure of C(Y)-algebra induced by the composition morphism C(Y) → C(X), is not single generated.