The theorem of this paper has appeared under stronger hypotheses and sometimes with weaker conclusions a number of times [l; 2; 3], and was known to Koch (in approximately the form of [2]) in 1959 (unpublished). Since it is a useful tool in the study of topological semigroups, and the proof here is simpler, and the theorem stronger than those mentioned, it seems advisable to have it in the literature. Lemma 1. Let X be a compact space and G a compact group of homeomorphisms of X. If X contains

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... fixed point under G, and X/G is connected, then X is connected. Proof. Let X = A\JB where A and B are compact, open subsets of