Existing in Mathlib · Topology

Baire Category Theorem

In a Baire space, the common intersection of any countable family of open dense subsets is still dense.

Baire spaces
countable intersections
open dense sets
set-indexed intersections
topological density

Exact theorem

Exact Mathlib statement

theorem dense_sInter_of_isOpen {X : Type*} [TopologicalSpace X] [BaireSpace X] {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable) (hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S)

The theorem at a glance

Baire category theorem at a glance

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

An ivory Baire category poster states the countable open-dense hypotheses and dense set-theoretic intersection conclusion above patterned layers whose thin gold common trace meets several nonempty open probe regions. — A countable family of open dense subsets of a Baire space has a dense set-theoretic intersection.

Statement structure

Statement and scope

Read the exact Mathlib declaration

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

Scientific figure illustrating: Overlaid open meshes share a gold trace that meets every probe region across one ambient field. — Overlaid open dense layers retain a distributed common trace across the ambient field.

Why it matters

A mathematical landmark

The Baire category theorem is a central bridge between local openness and global topological largeness. Mathlib's selected declaration isolates the abstract Baire-space core cleanly: countability and memberwise openness and density force the set-indexed common intersection to remain dense, with the empty family and enumeration step made explicit in the checked source.

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

This page indexes Mathlib's set-indexed sInter form of the Baire category theorem. In a topological space X carrying a BaireSpace instance, a countable family S : Set (Set X) whose every member is open and dense has dense set-theoretic intersection ⋂₀ S. The selected declaration concludes density only: it does not assert that the intersection is open or equal to X, add a nonempty-family hypothesis, or assume a metric, completeness, local compactness, or regularity.

The selected declaration concludes that ⋂₀ S is dense; it does not claim that this countable intersection is open or equal to X.
The family is a countable set S : Set (Set X), and the endpoint is the set-theoretic intersection ⋂₀ S; the page does not replace it with a finite-only or differently indexed statement.
The declaration does not require S to be nonempty; its proof handles the empty family separately before enumerating a nonempty countable family.
The declaration assumes BaireSpace X directly and does not itself assume or prove metric completeness, local compactness, or regularity.
ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

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