An application of combinatorial techniques to a topological problem

Abstract

The following statement is proved: Let X be a set having at most continuously many elements and f: X → X a mapping such that each iteration fn (n = 1, 2, …) has a unique fixed point. Then for every number c ∈ (0, 1) there exists a metric p on X such that the metric space (X, p) is separable and the mapping f is a.contraction with the Lipschitz constant c.