CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Samuel compactifications are unique up to the unique isomorphism fixing the original space

Statement

Under dependent choice and the ultrafilter lemma, two Samuel compactifications of the same separated uniform space are related by exactly one uniform isomorphism commuting with their canonical maps.

Facts & Assumptions

Given: Samuel compactifications $\eta_i:X\to S_i$ for $i=1,2$ of one separated uniform space, under dependent choice and the ultrafilter lemma.

[L1]

A Samuel compactification map is uniformly continuous from the Samuel uniformity; since that uniformity is coarser than the original one, it is also uniformly continuous from the original uniformity. A uniformly continuous map into a compact Hausdorff target then extends uniquely over a Samuel compactification (The Samuel completion and, when compactifying, the Samuel compactification, Samuel function pseudometrics generate a uniformity coarser than the original one, Under dependent choice and the ultrafilter lemma, uniformly continuous maps to compact Hausdorff spaces extend uniquely over the Samuel compactification).

[L2]

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

Proof

technique · direct

1.1

Apply [L1] to $\eta_2:X\to S_2$ and to $\eta_1:X\to S_1$ , obtaining uniformly continuous maps $F:S_1\to S_2$ and $G:S_2\to S_1$ with $F\eta_1=\eta_2$ and $G\eta_2=\eta_1$ .

2.1

The maps $GF$ and $\operatorname{id}_{S_1}$ agree on the dense set $\eta_1[X]$ , while $FG$ and $\operatorname{id}_{S_2}$ agree on $\eta_2[X]$ , so [L2] makes both composites identities.

L2step 1.1

3.1

Therefore $F$ and $G$ are inverse uniform isomorphisms, and uniqueness of $F$ is the uniqueness clause in [L1].

L1step 2.1∎

Depends on

Used by

Under dependent choice and the ultrafilter lemma, the Samuel compactification of the discrete natural numbers is beta N Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Zucker, Big Ramsey Degrees and Topological Dynamics (standard reference, not scraped)