LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The interior of an attached cell embeds openly in its closure

Statement

For an attached cell, its characteristic map is injective on ${\overset{˚}{D}}^{n}$ and maps this interior homeomorphically onto an open subset of its closed-cell image. The full disk map need not be injective.

Facts & Assumptions

Given: The adjunction quotient of Cell attachment by a characteristic map.

Proof

technique · direct

1.1given

The defining equivalence relation identifies only boundary points of $D^{n}$ with points of $X$ (and may identify boundary points with one another); no point of ${\overset{˚}{D}}^{n}$ is equivalent to a distinct point.

2.1step 1.1∎

The saturated open set ${\overset{˚}{D}}^{n}$ has quotient homeomorphic to its image, and its complement in the closed-cell image is the image of $S^{n - 1}$ . Thus the image is open relative to the attached space, while step 1.1 deliberately says nothing about the boundary.

Depends on

Cell attachment by a characteristic map

Used by

The closure of a CW cell need not be a closed ball Counterexample
Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
CW skeleta are closed and cells form a disjoint partition Proposition

Dependency tree · one level

1 result within one dependency step 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, Chapter 0 (standard reference, not scraped)