CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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The Hawaiian earring is not a CW complex with its punctured circles as cells

Statement refuted

The Hawaiian earring is a CW complex with the common tangency point as its $0$ -cell and each circle minus that point as an open $1$ -cell.

Counterexample

Given: The union in $R^{2}$ of circles $C_{n}$ of radius $1 / n$ centered at $(1 / n, 0)$ , tangent at the origin.

Proof technique: direct.

1.1given

For each $n \geq 1$ , take the explicit point $p_{n} = (1 / n, 1 / n)$ in the proposed open cell $C_{n} ∖ {0}$ , and put $S = {p_{n} : n \geq 1}$ . In the Euclidean subspace topology of the earring, $p_{n} \to 0$ , while $0 \notin S$ ; therefore $S$ is not closed.

2.1step 1.1given∎

If the proposed cells formed a CW complex, the closure of each $1$ -cell would be its circle $C_{n}$ , and the closure of the $0$ -cell would be ${0}$ . Each $C_{n} \cap S = {p_{n}}$ is closed in $C_{n}$ , and ${0} \cap S = \emptyset$ . The weak-topology condition of CW complex with closure finiteness and weak topology would therefore make $S$ closed in the earring, contradicting step 1.1. Hence the proposed cell structure is not a CW complex.

Depends on

CW complex with closure finiteness and weak topology

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

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