Papers With

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

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

Tomoki Uda

math.GN Aug 11, 2026 · v1
formalization topology
Read PDF arXiv abstract
TL;DR
A ZFC proof of Ishiki's conjecture that Met(D) is not completely metrizable for |D|=ℵ₁ was formalised in Lean 4 with Mathlib.
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.
Problem

Ishiki's Conjecture 5.1 asserts that for a discrete space D of cardinality ℵ₁, the space Met(D) of compatible metrics with the supremum-distance topology is not completely metrizable.

Approach

Working in ZFC, a set A⊆[0,1] of cardinality ℵ₁ that is not Fσ is constructed, so its complement is not Gδ and hence not completely metrizable. An explicit one-parameter family of metrics (d_t) on a discrete space D of size ℵ₁ is built, lying in Met(D) exactly when t is in the complement of A. The map t↦d_t is shown to be a bi-Lipschitz embedding of the complement onto a closed subset of Met(D), yielding a contradiction if Met(D) were completely metrizable. The proof was formalised in Lean 4 using Mathlib.

Results

The conjecture is proven in ZFC with no additional set-theoretic assumptions such as the continuum hypothesis, and the argument is machine-checked in Lean 4.