Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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A cell decomposition without the weak topology need not be a CW complex

Statement refuted

A closure-finite decomposition into characteristic cells is automatically a CW complex.

Counterexample

Given: The Hawaiian earring in its subspace topology, decomposed into its common vertex and the punctured constituent circles.

Proof technique: direct.

1.1

Each punctured circle is the image of the interior of a characteristic interval, and its closed circle meets only itself and the common vertex. Thus this is a closure-finite characteristic-cell decomposition.

given
2.1

Put the radius-1/n circle tangent at the origin with centre (1/n,0), and choose its far point xn=(2/n,0). The set {xn:n≥1} meets every closed cell in a closed set, but it is not closed in the Hawaiian earring since xn→0. Hence the topology is not the weak topology on these closed cells, so this decomposition is not a CW complex.

step 1.1∎

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