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CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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Stone–Čech compactifications are uniquely homeomorphic over the original space

Statement

Under the hypotheses of Under the ultrafilter lemma and dependent choice, the closure of the full evaluation image is the Stone–Čech compactification, two Stone–Čech compactifications (B,i)(B,i) and (B′,i′)(B',i') of XX are uniquely homeomorphic by a map u:B→B′u:B\to B' satisfying u∘i=i′u\circ i=i'.

Facts & Assumptions

Given: Stone–Čech compactifications (B,i)(B,i) and (B′,i′)(B',i') of XX under the stated choice hypotheses.

[L1]

The Stone–Čech property gives a unique continuous extension into every compact Hausdorff target (The Stone–Čech compactification by its compact-Hausdorff extension property).

[L2]

Continuous maps to a Hausdorff target agreeing on a dense subset are equal (Two continuous maps into a Hausdorff space that agree on a dense subset are equal).

Proof

technique · direct
1.1

Apply [L1] twice to obtain continuous u:B→B′u:B\to B' and v:B′→Bv:B'\to B with u∘i=i′u\circ i=i' and v∘i′=iv\circ i'=i.

L1
2.1

The maps v∘uv\circ u and id⁡B\operatorname{id}_B agree on i[X]i[X], which is dense in BB, so [L2] gives v∘u=id⁡Bv\circ u=\operatorname{id}_B. Similarly u∘v=id⁡B′u\circ v=\operatorname{id}_{B'}.

L2step 1.1
3.1

Hence uu is a homeomorphism over XX; its uniqueness is the uniqueness clause in [L1].

L1step 2.1∎

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