On Ishiki's Conjecture: $\mathrm{Met}(D)$ Is Not Completely Metrizable for $\lvert D\rvert=\aleph_1$
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Uda, Tomoki
Institution: Nanzan University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202315328