Proposition 12

(i) The subspace $i(X)$ is dense in $\hat{X}$. (ii) The graph of the equivalence relation $i(x) = i(x')$ is the intersection of the entourages of $X$. (iii) The uniform structure of $X$ is the inverse image under $i$ of that of $\hat{X}$ [or of that of the subspace $i(X)$]. (iv) The entourages of $i(X)$ are the images under $i \times i$ of the entourages of $X$, and the closures in $\hat{X} \times \hat{X}$ of the entourages of $i(X)$ form a fundamental system of entourages of $\hat{X}$.

(i) and (iii) have been proved in the course of the proof of Theorem 3; (iv) is a consequence of (i) and (iii) by virtue of general results proved earlier (§ 2, no. 4, Remark and Proposition 6). The relation

means by definition that $x$ and $x'$ have the same neighbourhood filter. But this implies, by definition, that $(x, x') \in V$ for every entourage $V$ of $X$, and the converse is obvious.