Existing in Mathlib · Topology

Tychonoff's Theorem

For each index i, choose a compact subset s i of a topological space X i. The set of all dependent functions x with x i inside s i for every i is compact in the product topology, even when the index type is infinite.

topological spaces
compact sets
product topology
arbitrary products

Exact theorem

Exact Mathlib statement

theorem isCompact_pi_infinite {ι : Type u₁} {X : ι → Type u₂} [(i : ι) → TopologicalSpace (X i)] {s : (i : ι) → Set (X i)} : (∀ i, IsCompact (s i)) → IsCompact {x | ∀ i, x i ∈ s i}

The theorem at a glance

Tychonoff's Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory theorem poster sends an open-ended index family into indexed topological coordinate spaces, marks one compact subset in each, and gathers the coordinate constraints into a compact dependent product subset. — Compact coordinate subsets over an arbitrary index type form a compact constrained dependent product, with no nonemptiness claim.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Five distinct compact coordinate regions project selected gold points into one enclosed dependent-tuple row, with two continuation ellipses indicating further coordinates. — Each coordinate is selected inside its compact constraint; the continuing family assembles into the constrained dependent product.

Why it matters

A mathematical landmark

Tychonoff's theorem is one of topology's central compactness principles. Its arbitrary-product scope is a decisive reason product topology is defined as it is.

Proof architecture

How ultrafilter limits prove product compactness

3 curated stages

The checked source rewrites compactness and product convergence in ultrafilter form, maps the ultrafilter through every coordinate evaluation, applies compactness in each factor, and then chooses the coordinate limits and reassembles them into convergence in the product topology.

These source-anchored stages explain the retained Mathlib proof route. They are generated explanations, not proof evidence or an extracted Lean proof-term dependency graph.

Product compactness becomes an ultrafilter convergence problem
Product compactness becomes an ultrafilter convergence problem
Mathlib rewrites compactness, product neighborhoods, and product membership so the proof begins with an ultrafilter containing the constrained product subset.
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Mathlib rewrites compactness, product neighborhoods, and product membership so the proof begins with an ultrafilter containing the constrained product subset.
f <= principal {x | forall i, x i in s i} implies a required product limit
`isCompact_iff_ultrafilter_le_nhds`, `nhds_pi`, `le_pi`, and `le_principal_iff` transform the compactness goal into a convergence problem for an arbitrary ultrafilter lying over the dependent product subset. This reformulation introduces no coordinate witnesses yet.
Every coordinate projection acquires a compact limit
Every coordinate projection acquires a compact limit
Mapping the ultrafilter through each evaluation map preserves membership in the corresponding compact set, so coordinate compactness provides a limit point there.
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Mapping the ultrafilter through each evaluation map preserves membership in the corresponding compact set, so coordinate compactness provides a limit point there.
exists x, x in s i and Tendsto (Function.eval i) f (nhds x)
For each index `i`, Mathlib maps the ultrafilter through `Function.eval i`. Membership of the constrained product implies that the mapped ultrafilter contains `s i`; compactness of `s i` then supplies a point `x i` in that set and convergence of the projected filter to it.
Coordinate witnesses assemble into a product-filter limit
Coordinate witnesses assemble into a product-filter limit
Chosen coordinate limits satisfy xᵢ ∈ sᵢ; reassembly yields the source-faithful product-filter relation f ≤ 𝓝 x.
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Chosen coordinate limits satisfy xᵢ ∈ sᵢ; reassembly yields the source-faithful product-filter relation f ≤ 𝓝 x.
x i ∈ s i for every i, and f ≤ 𝓝 x
A single charcoal source object at left sends cobalt paths through four representative evaluation apertures, while dotted continuation marks keep the coordinate family open-ended. Every visible path ends at exactly one antique-gold point inside a forest-green compact region. Dotted gold paths gather the coordinate witnesses into one product point at right, enclosed by nested green neighborhood contours. The two visible formulas record coordinate membership xᵢ ∈ sᵢ and the type-correct product-filter conclusion f ≤ 𝓝 x.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

The selected Mathlib declaration proves compactness of the dependent product subset {x | ∀ i, x i ∈ s i}. It does not state a finite-product-only result, and the page does not hide the family of topology instances or identify explanatory artwork with proof evidence.

Proof Atlas did not originate Tychonoff's theorem or Mathlib's declaration.
The local evidence checks this exact upstream declaration at one pinned Mathlib commit.
The page does not independently classify the theorem's foundational choice principles beyond the recorded axiom report.

Source and local evidence

Where the theorem comes from

Existing declaration: isCompact_pi_infinite in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.tychonoff-theorem.v001
Source: Open the pinned upstream reference