Semigroup structures for families of functions, I. Some Homomorphism Theorems

Abstract

This is the first of several papers which grew out of an attempt to provideC (X, Y), the family of all continuous functions mapping a topological spaceXinto a topological spaceY, with an algebraic structure. In the eventYhas an algebraic structure with which the topological structure is compatible, pointwise operations can be defined onC (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ringC (X, R)of all continuous, real-valued functions defined on X [3]. Now, one can provideC(X, Y)with an algebraic structure even in the absence of an algebraic structure onY. In fact, each continuous function fromYintoXdetermines, in a natural way, a semigroup structure forC(X, Y). To see this, let ƒ be any continuous function fromYintoXand for ƒ andginC(X, Y), define ƒgbyeachxinX.