The Samuel completion and, when compactifying, the Samuel compactification

A Samuel completion of $(X,\mathcal U)$ is a Hausdorff completion

$$\eta:(X,\mathcal U_S)\longrightarrow S(X)$$

of its Samuel uniformity in the sense of A Hausdorff completion of a uniform space and its canonical dense map. Such a completion exists by 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, but its canonical map need not be injective.

Regard $X$ with its original induced topology. A Samuel completion $\eta:(X,\mathcal U_S)\to S(X)$ is a Samuel compactification if and only if the same map $\eta:X\to S(X)$ makes $(S(X),\eta)$ a compactification in the sense of A Hausdorff compactification as a dense embedding into a compact Hausdorff space. In particular, this requires $S(X)$ to be compact Hausdorff and $\eta$ to be an embedding with dense image; it is not used merely for a Hausdorff completion of a nonseparated uniform space.