Theorems · Theorem · general topology

ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints

• 1000+ list: Stone–Weierstrass theorem

∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)),
  A.SeparatesPoints → A.topologicalClosure = ⊤

The Stone-Weierstrass Approximation Theorem, that a subalgebra A of C(X, ℝ), where X is a compact topological space, is dense if it separates points.

Defined in

Mathlib.Topology.ContinuousMap.StoneWeierstrass

Cited by

3 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceCompactSpace