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If a map A: X-*X: x-*Ax in a complete metric space (X, d) is such that, for some c<1, d(Ax, Ay) cd(x, y) holds for all x, yEX, then the equation Ax = x has in X a unique solution. The usefulness of this "principle of contraction mappings" in analysis has been well illustrated by Kolmogorov and Fomin in [2, pp. 43-51]. Presented with an elegant little theorem of great applicability such as this, one is moved to wonder, how far can it be pushed? Can the concept of "contraction" be made meaningful in more general spaces and, if so, does the quoted fixpoint theorem have a significant generalization? For regular developable spaces an affirmative answer has been provided by Dr. J. Mathews [3], to whom I owe my interest in this question. The present paper offers a fixpoint theorem for well-chained spaces somewhat more general than those furnished with uniformities. The definitions proposed in the latter half of ?1 and the lemmata leading to Theorem 2 of ?2 are formulated in as general a manner as seemed compatible with naturalness. It may be that they are of interest independently of the particular fixpoint theorem presented here. (Some of the definitions differ from those tentatively proposed in [1].)

Fixpoint theorem for contractions of a well-chained topological space