A Characterization of Distal and Point-distal Minimal Transformation Groups1

If G is a locally compact topological group, let BC{G) denote the set of real-valued, bounded, uniformly continuous functions on G with the compact-open topology. Using the fact that the distal (weakly distal) functions are the elements of BC(G) whose orbit closures are compact distal (point-distal) minimal sets, we can characterize compact distal and point-distal minimal transformation groups. Let (X, G, ) be a right transformation group where A'is compact Hausdorff and minimal under G. Then X is a compact distal (point-distal) minimal set if and only if there is a point xeX such that for any homomorphism h: X-*BC(G), h(x) is a right distal (weakly distal) function.