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Let X be a metric space and let G(X) be the space of homeomorphisms of X with the compact open topology. Theorem. If G(X) is locally compact, then it is zero-dimensional. Some examples are also given of metric spaces X for which C(X) is compact.doi:10.2307/2038149 fatcat:7a4yzut3p5d5vaahw6wbklhsly