Existing in Mathlib · Topology

Urysohn's Lemma

Two disjoint closed sets in a normal topological space can be separated continuously: a function takes the value zero on one set, one on the other, and stays between those values everywhere.

normal spaces
disjoint closed sets
continuous functions
topological separation
unit interval

Exact theorem

Exact Mathlib statement

theorem exists_continuous_zero_one_of_isClosed {X : Type*} [TopologicalSpace X] [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) : ∃ f : C(X, ℝ), Set.EqOn f 0 s ∧ Set.EqOn f 1 t ∧ ∀ x, f x ∈ Set.Icc (0 : ℝ) 1

The theorem at a glance

Urysohn's Lemma at a glance

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

An ivory topology poster places two disjoint closed regions at opposite ends of a continuous zero-to-one height field built from nested open and closed neighborhoods. — Normality supplies a continuous [0,1]-valued function that is zero on one closed set and one on the other.

Statement structure

From hypotheses to conclusion

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

Two disjoint closed islands sit inside a normal topological landscape while nested contour bands interpolate smoothly from an ivory zero shore to a gold one shore. — A continuous [0,1]-valued field separates the two closed sets exactly.

Why it matters

A mathematical landmark

Urysohn's lemma is a central separation theorem and a gateway from abstract topology to continuous real-valued functions. It underlies metrization, embedding, and extension arguments.

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 Urysohn lemma for two disjoint closed subsets of a normal topological space. It produces a continuous real-valued function equal to zero on the first set, equal to one on the second, and valued in the closed interval [0,1]. It does not assert that the sets are nonempty, that the separator is unique, that it is a homeomorphism, or that the space is metrizable.

Normality is an explicit hypothesis; this declaration does not prove the space normal.
The separator is continuous and [0,1]-valued but is not claimed to be unique, injective, or a homeomorphism.
The result separates two closed sets; it does not assert a metric or numerical distance between them.

Source and local evidence

Where the theorem comes from

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