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A topological space X supports a nontrivial Haar system if and only if there is a continuous one-to-one map of X into the unit circle. This gives an elementary proof of Mairhuber's classical theorem for compact X. It answers a 1960 question asked by R. R. Phelps when X is locally compact.doi:10.1090/s0002-9939-1985-0792284-7 fatcat:7c4siauq3ff4rehixilbp3jrby