On a nonempty set, entourage uniformities and uniform-cover structures determine one another

From an entourage $E$, form ${\mathcal{C}}_E=\{E[x]:x\in X\}$. Choose a symmetric entourage $D$ with $D^{\circ 3}\subseteq E$. If $D[y]\cap D[x]\mathrm{\ne }\mathrm{\varnothing }$ and $z\in D[y]$, symmetry and a point in the intersection give $(x,z)\in D^{\circ 3}\subseteq E$. Hence the star of $D[x]$ in ${\mathcal{C}}_D$ lies in $E[x]$, so ${\mathcal{C}}_D$ star-refines ${\mathcal{C}}_E$.

From a uniform cover $\mathcal{V}$, form $E_{\mathcal{V}}={\bigcup }_{V\in \mathcal{V}}V\times V$. It contains the nonempty diagonal, so it is nonempty. A star-refinement $\mathcal{W}$ has $E_{\mathcal{W}}\circ E_{\mathcal{W}}\subseteq E_{\mathcal{V}}$, while common refinements and coarsenings give the remaining filter axioms.

Declare a cover uniform when it is coarser than some ${\mathcal{C}}_E$. Intersections of entourages give common refinements, enlargement of an entourage gives coarsening, and step 1.1 gives star-refinements. Hence these covers satisfy the uniform-cover axioms.

Start with an entourage uniformity. For symmetric $D$, $D\subseteq E_{{\mathcal{C}}_D}\subseteq D^{-1}\circ D=D^{\circ 2}\mathrm{.}$ The first inclusion uses the diagonal, and the second follows because two points in one $D$-ball are $D^{-1}\circ D$-related. Taking a symmetric square root inside any prescribed entourage shows that the recovered entourage filter is exactly the original one.

Start instead with a uniform-cover structure. The $E_{\mathcal{V}}$-ball at $x$ is $E_{\mathcal{V}}[x]=\mathrm{St}(x,\mathcal{V}),$ the union of the members of $\mathcal{V}$ containing $x$. Thus $\mathcal{V}$ refines ${\mathcal{C}}_{E_{\mathcal{V}}}$, so the latter is uniform by coarsening. Conversely, if $\mathcal{W}$ star-refines $\mathcal{V}$, then for any $x$ and any $W_0\in \mathcal{W}$ containing $x$, $\mathrm{St}(x,\mathcal{W})\subseteq \mathrm{St}(W_0,\mathcal{W})$, which lies in some member of $\mathcal{V}$. Therefore ${\mathcal{C}}_{E_{\mathcal{W}}}$ refines $\mathcal{V}$. The recovered cover structure is exactly the original one.

Steps 2.2 and 2.3 prove that the two constructions are mutually inverse at the level of generated structures.