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.

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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

1.1

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 $N$ .

2.1

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.

step 1.1L1

3.1

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

step 2.1∎

Depends on

Used by

Dependency tree · two levels

11 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. Robbin, Partitions of Unity (standard reference, not scraped)
S. Semmes, Topology notes, Sections 5.13–5.14 (Rice University) (standard reference, not scraped)