CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A point-separating nowhere-vanishing real function algebra is uniformly dense

Statement

Let $X$ be a compact Hausdorff space and let $A \subseteq C (X, R)$ be a point-separating nowhere-vanishing real function algebra. Then $A$ is uniformly dense in $C (X, R)$ .

Facts & Assumptions

Given: A compact Hausdorff space $X$ and a point-separating nowhere-vanishing real function algebra $A \subseteq C (X, R)$ .

[L1]

A point-separating real function algebra either has full uniform closure, or all its members vanish at one fixed point and its closure is exactly the functions vanishing there; the empty space lies in the full-closure alternative (A separating real function algebra is dense or its closure consists exactly of the functions vanishing at one point).

Proof

technique · direct

1.1

If $X = \emptyset$ , [L1] gives the full-closure conclusion directly.

2.1

If $X \neq \emptyset$ , the proper alternative in [L1] would give a point $x_{0}$ at which every member of $A$ vanishes, contradicting nowhere-vanishing at $x_{0}$ ; therefore the full-closure alternative holds and $A$ is uniformly dense.

L1given∎

Depends on

A separating real function algebra is dense or its closure consists exactly of the functions vanishing at one point

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, consequence of Theorem 1.26 (standard reference, not scraped)