Every Metric Space Is Separable in Function Realizability

Explore a lecture from the Hausdorff Trimester Program on Constructive Mathematics that delves into function realizability topos and its implications for metric spaces and objects with decidable equality. Learn how every metric space is separable and every object with decidable equality is countable in this context. Discover the broader applications to synthetic topology, where T0-spaces are shown to be separable and discrete spaces are countable. Examine the consequences for intuitionistic logic, including the inability to prove the existence of non-separable metric spaces or uncountable sets with decidable equality, even when considering principles validated by function realizability such as Dependent and Function choice, Markov's principle, and Brouwer's continuity and fan principles.