Every metric space is separable in function realizability

Autor: Andrej Bauer, Andrew Swan
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Zdroj: Logical Methods in Computer Science, Vol Volume 15, Issue 2 (2019)
Druh dokumentu: article
ISSN: 1860-5974
DOI: 10.23638/LMCS-15(2:14)2019
Popis: We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every discrete space is countable. It follows that intuitionistic logic does not show the existence of a non-separable metric space, or an uncountable set with decidable equality, even if we assume principles that are validated by function realizability, such as Dependent and Function choice, Markov's principle, and Brouwer's continuity and fan principles.
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