Properties of standard maps

Abstract:

Let X and Y be compact metric spaces. Let $S(X,Y)$ denote the collection of standard maps of X onto Y. We establish that $S(C,Y)$ is a dense subset of $C(C,Y)$, where C is the Cantor set. If f is a standard map and $G(f,Y)\{ A(f,Y)\}$ denotes the subgroup of $H(X)$ which preserves {interchanges} the point-inverses of f, then there is a continuous homomorphism of $A(f,Y)$ into $H(Y)$ with kernel $G(f,Y)$. We also show that $G(f,Y)$ and $A(f,Y)$ are closed subsets of $H(X)$.