A Systematic Generalization Procedure for Fixed-Point Theorems

Abstract

The progress of mathematics often follows a standard path: the discovery of a new theorem, followed by a systematic exploration of that theorem. Two standard ways of exploring theorems are by weakening the hypotheses and strengthening the conclusion. This paper discusses another way to explore theorems through the sharing of hypotheses, and develops this idea in the context of three classical fixedpoint theorems: the Banach Contraction Principle, the Tarski Fixed-Point Theorem for complete lattices and the Brouwer Fixed-Point Theorem for solid n-spheres.