ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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 compact set in an infinite non locally finite realization still meets finitely many cells

Example

Let $K$ have vertices $o, v_{1}, v_{2}, \dots$ , edges ${o, v_{n}}$ and their faces, with weak realization topology. Every compact subset lies in finitely many edges although the vertex $o$ has infinitely many incident edges. An explicit compact subset is $C_{0} = {(1 - t) o + t v_{1} : 0 \leq t \leq 1} \cup {(1 - t) o + t v_{2} : 0 \leq t \leq 1 / 2} .$ In contrast the set $Q = {(1 - 1 / n) o + (1 / n) v_{n} : n \geq 1}$ is closed discrete and not compact. For this countably indexed fan the conclusion for every compact subset has the choice-free proof below.

Source locators

Proposition A.1 p.520, simplicial specialization.

Facts & Assumptions

[F1]

Stars and weak topology use simplexwise coordinates. Open and closed stars in a subdivision.

Verification

Given: The countably indexed fan with weak topology and, when proving the universal claim, a compact subset $C$ .

1.1

The two segments defining $C_{0}$ are continuous images of compact closed real intervals, so their finite union is compact. Its positive supports are ${o}$ , ${v_{1}}$ , ${o, v_{1}}$ and ${o, v_{2}}$ , exactly four open simplices in two edges. Any neighbourhood of $o$ intersects each edge in a relative neighbourhood of its endpoint, hence meets the interior of every edge. Thus $o$ is not locally finite.

1.2

Each closed edge meets $Q$ in exactly one point, and the central vertex is not in $Q$ . Every subset of $Q$ therefore has finite closed traces on every simplex and is weakly closed. In particular $Q$ is closed and discrete, and its singleton cover has no finite subcover. The numerical parameter $1 / n$ tending to zero does not imply convergence to $o$ in the weak topology: $∣ K ∣ ∖ Q$ is an open neighbourhood of $o$ missing all these points.

1.3

For a compact nonempty $C$ , the coordinate $x \mapsto x_{v_{n}}$ is continuous because it is continuous on every simplex. Let $m_{n} = \sup {x_{v_{n}} : x \in C} \in [0, 1]$ . This supremum is attained: otherwise the sets ${x \in C : x_{v_{n}} < m_{n} - 1 / k}$ for positive integers $k$ would cover $C$ , and a finite subcover would bound all coordinates by $m_{n} - 1 / k_{0}$ for its largest index $k_{0}$ , contradicting the supremum. If $m_{n} > 0$ , there is exactly one point of the entire fan with coordinate $x_{v_{n}} = m_{n}$ , namely $p_{n} = (1 - m_{n}) o + m_{n} v_{n}$ , so this point belongs to $C$ . These points are canonical and require no countable selections.

given

2.1

If infinitely many $m_{n}$ are positive, the corresponding $p_{n}$ form an infinite set with at most one point on each edge, every subset of which is weakly closed just as for $Q$ . It is therefore closed in compact $C$ , and is compact (adjoin its open complement to any cover to see this directly). Yet it is discrete with an infinite singleton cover, a contradiction. Only finitely many $m_{n}$ are positive, so $C$ lies in their edges and possibly $o$ ; if none are positive use the first edge to contain $o$ . For empty $C$ no edge is needed.

step 1.2step 1.3∎

Remarks

Under Countable Choice the general result Compact subsets of an arbitrary simplicial realization meet finitely many open simplices supplies the compact-support conclusion immediately. The specialized maximum-coordinate argument above proves this countably indexed example without adding that hypothesis. The supplied instance $C_{0}$ and the noncompact set $Q$ are explicit, rather than unspecified selections of points on the edges.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)