A fixed point theorem for o‐minimal structures | Zendy

Wong KamChau | Zendy

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A fixed point theorem for o‐minimal structures

Author(s) -

Wong KamChau

Publication year - 2003

Publication title -

mathematical logic quarterly

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.473

H-Index - 28

eISSN - 1521-3870

pISSN - 0942-5616

DOI - 10.1002/malq.200310065

Subject(s) - mathematics , lemma (botany) , simplex , fixed point , fixed point theorem , triangulation , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , geometry , ecology , poaceae , biology

We prove a definable analogue to Brouwer's Fixed Point Theorem for o‐minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o‐minimal functions, with an application of Sperner's Lemma. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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