Tietze-Urysohn theorem fails in constructive mathematics

Jun He; Xizhe Jiang; Ruizhe Wan; Yixiang Weng

doi:10.1117/12.2628144

22 April 2022 Tietze-Urysohn theorem fails in constructive mathematics

Jun He, Xizhe Jiang, Ruizhe Wan, Yixiang Weng

Proceedings Volume 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021); 121632W (2022) https://doi.org/10.1117/12.2628144
Event: International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 2021, Nanjing, China

Abstract

In this paper, we prove that the classical Tietze-Urysohn theorem in analysis fails in constructive metric spaces. We find two examples of constructive metric spaces. In each of them, based on the existence of an un-extendable computable function, we create closed sets 𝐴 and 𝐵 such that there does not exist a constructive function 𝑓: 𝑋(𝑜𝑟 𝑌) → [0,1] satisfying 𝑓⁻¹(0) = 𝐴 and 𝑓⁻¹(1) = 𝐵.

Citation Download Citation

Jun He, Xizhe Jiang, Ruizhe Wan, and Yixiang Weng "Tietze-Urysohn theorem fails in constructive mathematics", Proc. SPIE 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 121632W (22 April 2022); https://doi.org/10.1117/12.2628144

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