Theorems · Theorem · general topology

ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints

∀ {X : Type u_1} [inst : TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)),
  A.SeparatesPoints → ∀ (f : X → ℝ), Continuous f → ∀ (ε : ℝ), 0 < ε → ∃ g, ∀ (x : X), ‖↑g x - f x‖ < ε

An alternative statement of the Stone-Weierstrass theorem, for those who like their epsilons and don't like bundled continuous functions. If A is a subalgebra of C(X, ℝ) which separates points (and X is compact), every real-valued continuous function on X is within any ε > 0 of some element of A.

Defined in

Mathlib.Topology.ContinuousMap.StoneWeierstrass

Cited by

1 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceCompactSpace