  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.
