Equivalence_of_Definitions_of_Connected_Topological_Space

⚠️ 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 $T = \struct {S, \tau}$ be a topological space.

Why this node was retired

The posted statement is

theorem Equivalence_of_Definitions_of_Connected_Topological_Space {α : Type _} [TopologicalSpace α] : ConnectedSpace α ↔ ¬ ∃ (U V : Set α), IsOpen U ∧ IsOpen V ∧ U ∪ V = Set.univ ∧ (U ∩ V = ∅) ∧ U.Nonempty ∧ V.Nonempty := by sorry

ConnectedSpace includes nonemptiness, but absence of a separation allows the empty space.

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

Proposed corrected statement

Add Nonempty α, or replace ConnectedSpace by PreconnectedSpace, in the stated equivalence with absence of two disjoint nonempty open sets covering the space.

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.