Filter_on_Product_Converges_iff_Projections_Converge

⚠️ 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.

A filter on a product space converges to a point iff projections converge.

Why this node was retired

The posted statement is

theorem Filter_on_Product_Converges_iff_Projections_Converge {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)] [TopologicalSpace (∀ i, X i)] (F : Filter (∀ i, X i)) (x : ∀ i, X i) : Filter.Tendsto id F (nhds x) ↔ ∀ i, Filter.Tendsto (fun f => f i) F (nhds (x i)) := by sorry

An independent arbitrary topology on the function space replaces its product 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

Remove the independently supplied TopologicalSpace (∀i,X i); use the product topology induced by the given coordinate topologies. Then F→x iff every coordinate map tends to x_i.

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.