Nonexistence of Brouwer’s fixed-point theorem in one-dimensional constructive analysis

Wenshi Zhao; Liran Li; Zixuan Qiao; Hongting Zhu

doi:10.1117/12.2628159

22 April 2022 Nonexistence of Brouwer’s fixed-point theorem in one-dimensional constructive analysis

Wenshi Zhao, Liran Li, Zixuan Qiao, Hongting Zhu

Proceedings Volume 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021); 121630H (2022) https://doi.org/10.1117/12.2628159
Event: International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 2021, Nanjing, China

Abstract

This paper reviews the Brouwer’s fixed-point theorem under the lens of constructive mathematics. Specifically, it presents a proof for the following statement: there is no analog of Brouwer’s fixed-point theorem in one-dimension in constructive mathematics. The paper starts by introducing some basic concepts of constructive mathematics and the Brouwer’s fixed-point theorem in classical topology. It then gives a counterexample to Brouwer’s fixed-point theorem on the interval [0,1] with idea that there exists a non-extendable computable function in constructive mathematics. The result of this work provides further insights on the characteristics of constructive mathematics.

Citation Download Citation

Wenshi Zhao, Liran Li, Zixuan Qiao, and Hongting Zhu "Nonexistence of Brouwer’s fixed-point theorem in one-dimensional constructive analysis", Proc. SPIE 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 121630H (22 April 2022); https://doi.org/10.1117/12.2628159

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