CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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 $η_{i} : X \to S_{i}$ 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 $η_{2} : X \to S_{2}$ and to $η_{1} : X \to S_{1}$ , obtaining uniformly continuous maps $F : S_{1} \to S_{2}$ and $G : S_{2} \to S_{1}$ with $F η_{1} = η_{2}$ and $G η_{2} = η_{1}$ .

2.1

The maps $G F$ and ${id}_{S_{1}}$ agree on the dense set $η_{1} [X]$ , while $F G$ and ${id}_{S_{2}}$ agree on $η_{2} [X]$ , 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 · two levels

25 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

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