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Brouwer’s approximate fixed-point theorem is equivalent to Brouwer’s fan theorem. (English) Zbl 1167.03040

Lindström, Sten (ed.) et al., Logicism, intuitionism, and formalism. What has become of them? Originated from the conference and the symposium on constructive mathematics, Uppsala, Sweden, August 2004. Dordrecht: Springer (ISBN 978-1-4020-8925-1/hbk; 978-1-4020-8926-8/e-book). Synthese Library 341, 277-299 (2009).
This article establishes a relationship between an axiom and a theorem which are both named after Brouwer, namely the fan theorem and the fixed-point theorem. After giving an exact definition of continuous functions in a formal system, the author proves results like the equivalence of the fan theorem to the statement “every continuous function from the unit square into itself has approximate fixpoints”. Everyone interested in constructive mathematics will benefit from reading this paper.
For the entire collection see [Zbl 1154.03003].

MSC:

03F60 Constructive and recursive analysis
47H10 Fixed-point theorems
47S30 Constructive operator theory
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