Under the ultrafilter lemma the Samuel completion is compact, and under dependent choice plus the ultrafilter lemma it compactifies every separated uniform space

Given: A uniform space $(X,\mathcal{U})$, a Samuel completion $\eta :(X,{\mathcal{U}}_S)\to S(X)$, the ultrafilter lemma, and, for the final assertion, dependent choice and separatedness of $\mathcal{U}$.

The Samuel uniformity is totally bounded (The Samuel uniformity is totally bounded).

A Hausdorff completion has complete separated target, dense image, and uniformly continuous canonical map; its canonical map is a uniform embedding exactly for separated source uniformity (A Hausdorff completion of a uniform space and its canonical dense map, Every uniform space has a Hausdorff completion with dense canonical image, and the canonical map is a uniform embedding exactly when the original uniformity is separated, The Samuel completion and, when compactifying, the Samuel compactification).

A dense uniformly continuous image of a totally bounded uniform space is totally bounded (Total boundedness passes to a uniform space with a dense uniformly continuous image).

Under the ultrafilter lemma, every complete totally bounded uniform space is compact (Assuming the ultrafilter lemma, every complete and totally bounded uniform space is compact).

Under dependent choice the Samuel and original induced topologies agree; separatedness is equivalent to Hausdorffness of the induced topology, and a separated uniformizable topology is Tychonoff (Assuming dependent choice, the Samuel uniformity induces the original topology, A uniformity is separated if and only if its induced topology is Hausdorff, Assuming dependent choice, a nonempty topological space is separated-uniformizable if and only if it is Tychonoff).