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 $D^{\circ3}\subseteq E$. If $D[y]\cap D[x]\ne\varnothing$ and $z\in D[y]$, symmetry and a point in the intersection give $(x,z)\in D^{\circ3}\subseteq E$. 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 $D$, $D\subseteq E_{\mathcal C_D}\subseteq D^{-1}\circ D=D^{\circ2}.$ 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 $E_{\mathcal V}[x]=\operatorname{St}(x,\mathcal V),$ 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 $x$, $\operatorname{St}(x,\mathcal W)\subseteq\operatorname{St}(W_0,\mathcal W)$, 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.