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arXiv · 2608.11023

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

Tomoki Uda
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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 $F_σ$ and embedding its complement as a closed subspace of $\mathrm{Met}(D)$. The proof has also been formalised in Lean 4.

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Tomoki Uda. 2026-08-11. On Ishiki's Conjecture: $\mathrm{Met}(D)$ Is Not Completely Metrizable for $\lvert D\rvert=\aleph_1$. https://arxiv.org/abs/2608.11023

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