CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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 $(B, i)$ and $(B^{'}, i^{'})$ of $X$ are uniquely homeomorphic by a map $u : B \to B^{'}$ satisfying $u \circ i = i^{'}$ .

Facts & Assumptions

Given: Stone–Čech compactifications $(B, i)$ and $(B^{'}, i^{'})$ of $X$ 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 $u : B \to B^{'}$ and $v : B^{'} \to B$ with $u \circ i = i^{'}$ and $v \circ i^{'} = i$ .

2.1

The maps $v \circ u$ and ${id}_{B}$ agree on $i [X]$ , which is dense in $B$ , so [L2] gives $v \circ u = {id}_{B}$ . Similarly $u \circ v = {id}_{B^{'}}$ .

L2step 1.1

3.1

Hence $u$ is a homeomorphism over $X$ ; its uniqueness is the uniqueness clause in [L1].

L1step 2.1∎

Depends on

Used by

Under the ultrafilter lemma and Dependent Choice, a Tychonoff space is G_δ in some Hausdorff compactification exactly when it is G_δ in every one Theorem

Dependency tree · two levels

15 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

Stacks Project, Stone–Čech compactification (standard reference, not scraped)