Indiscrete_Topology_is_not_Metrizable

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $S$ be a set with more than one element. The indiscrete topology on $S$ is not metrizable.

Why this node was retired

The posted statement is

theorem Indiscrete_Topology_is_not_Metrizable {α : Type _} [Nontrivial α] : ¬ ∃ (_ : MetricSpace α), ∀ (s : Set α), IsOpen s := by sorry

Every subset being open characterizes the discrete topology, not the indiscrete topology.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For a nontrivial type α, there is no metric whose induced topology is the indiscrete topology; require ∀s, IsOpen s ↔ s=∅ ∨ s=Set.univ, or equality of the induced topology with ⊤.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.