Uniform space in the uniform-cover formulation

For a cover $\mathcal V$ of $X$ and $A\subseteq X$, write $\operatorname{St}(A,\mathcal V)$ for the union of the members of $\mathcal V$ meeting $A$. A cover $\mathcal V$ star-refines $\mathcal W$ if for every $V\in\mathcal V$, $\operatorname{St}(V,\mathcal V)$ is contained in some member of $\mathcal W$.

A uniform-cover structure is a nonempty family $\mathfrak C$ of covers of $X$ such that a cover refined by a member of $\mathfrak C$ belongs to $\mathfrak C$, any two members have a common refinement in $\mathfrak C$, and every member has a star-refinement in $\mathfrak C$. Its members are uniform covers. When $X\ne\varnothing$, the topology it induces and its equivalence with entourages are proved in On a nonempty set, entourage uniformities and uniform-cover structures determine one another.