On Ishiki's Conjecture: $\mathrm{Met}(D)$ Is Not Completely Metrizable for $\lvert D\rvert=\aleph_1$

Abstract

For a discrete topological space $D$, let $\mathrm{Met}(D)$ denote the set of metrics on $D$ that are compatible with the discrete topology, equipped with the topology induced by the supremum distance. Ishiki's Conjecture 5.1 asserts that $\mathrm{Met}(D)$ is not completely metrizable when $\lvert D\rvert = \aleph_1$. We prove this in ZFC by constructing a set $A \subseteq [0,1]$ of cardinality $\aleph_1$ that is not $F_\sigma$ and embedding its complement as a closed subspace of $\mathrm{Met}(D)$. The proof has also been formalised in Lean 4.