LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 $\{\operatorname{coz}(f_s)\}_{s\in S}$ 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 $\operatorname{coz}(f_{s_1}),\ldots,\operatorname{coz}(f_{s_n})$ ; every $f_s$ with $s\notin\{s_1,\ldots,s_n\}$ vanishes on $N$ .

2.1

Thus at every point of $N$ the displayed pointwise sum equals the finite sum $f_{s_1}+\cdots+f_{s_n}$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 12 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

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