LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A locally finite family of continuous nonnegative functions has a continuous pointwise sum

Statement

Let ${f_{s} : X \to [0, \infty)}_{s \in S}$ be continuous and suppose that ${coz (f_{s})}_{s \in S}$ is locally finite. Then $f (x) : = \sum_{s \in S} f_{s} (x)$ is a well-defined continuous map $X \to [0, \infty)$ .

Facts & Assumptions

Given: A locally finite family of cozero sets of continuous nonnegative functions on $X$ .

[F1]

At every point, a locally finite family has a neighbourhood meeting only finitely many members (Locally finite partitions of unity and subordination to an open cover).

[L1]

A finite sum of continuous real-valued maps is continuous (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined).

Proof

technique · direct

1.1F1

Fix $x \in X$ and a neighbourhood $N$ meeting only $coz (f_{s_{1}}), \dots, coz (f_{s_{n}})$ ; every $f_{s}$ with $s \notin {s_{1}, \dots, s_{n}}$ vanishes on $N$ .

2.1step 1.1L1

Thus at every point of $N$ the displayed pointwise sum equals the finite sum $f_{s_{1}} + \dots + f_{s_{n}}$ , so it is well defined and agrees on $N$ with a continuous function.

3.1step 2.1∎

Since every point has such a neighbourhood $N$ , the pointwise sum is continuous on $X$ and is nonnegative.

A locally finite family of continuous nonnegative functions has a continuous pointwise sum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources