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We prove a type of converse of the Banach contraction mapping theorem for metric spaces: if X is a T1 topological space and f: X -> X is a function with the unique fixed point a such that fn(x) converges to a for each x is a member of X, then there exists a distance function d on X such that f is a contraction on the complete ultrametric space (X,d) with contractivity factor 1/2. We explore properties of the resulting space (X,d).