LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A unital separating real function lattice interpolates arbitrary values at two distinct points

Statement

Let $X$ be a compact Hausdorff space and let $L \subseteq C (X, R)$ be a unital point-separating real vector sublattice (Unital point-separating real vector sublattices of $C (X, R)$ ). If $x, y \in X$ are distinct and $α, β \in R$ , then there is $h \in L$ with $h (x) = α and h (y) = β .$

Facts & Assumptions

Given: A unital point-separating real vector sublattice $L \subseteq C (X, R)$ , distinct points $x, y \in X$ , and prescribed values $α, β \in R$ .

[L1]

Point separation supplies $g \in L$ with $g (x) \neq g (y)$ , while the vector-space and unital clauses keep every affine combination $a g + b$ in $L$ (Unital point-separating real vector sublattices of $C (X, R)$ ).

Proof

technique · direct

1.1L1choose

By [L1], choose $g \in L$ with $g (x) \neq g (y)$ and put $a : = (β - α) / (g (y) - g (x))$ and $b : = α - a g (x)$ ; the denominator is nonzero because $g$ separates $x$ and $y$ .

2.1step 1.1L1algebra∎

The affine combination $h : = a g + b$ belongs to $L$ by [L1], and substitution gives $h (x) = a g (x) + α - a g (x) = α$ and $h (y) = α + a (g (y) - g (x)) = β$ .

Depends on

Unital point-separating real vector sublattices of $C(X,\mathbb R)$

Used by

Lattice Stone–Weierstrass theorem on a compact Hausdorff space Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. M. Erdman, A Companion to Real Analysis, Proposition 21.2.5 (standard reference, not scraped)
E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, Lemma 1.27 (standard reference, not scraped)