Kelley in his book General Topology introduces a notion of even cover. A cover $\mathcal{U}$ of a topological space $X$ is even if there exists a neighborhood $\mathcal{V} \subset X \times X$ of the diagonal such that for each $x \in X$, the set $\mathcal{V}[x] \subset U$ for some $U \in \mathcal{U}$. (Here $V[x]$ means all $y \in X$ such that $(x, y) \in V$.) He uses it in the way that later sources, like Engelking, would use star-refinement. For example, it appears in his characterization of paracompactness: every open cover of a paracompact space is even, and the converse holds for a regular space.
I can see that they are related (it's basically an entourage, while a star-refinement is a uniform cover in the sense of Tukey), but I'm not clear of the exact relationship. I did some quick Googling, and it looks like the idea of even covers didn't catch on.
What is the relationship between the idea of an even cover and a star-refinement? I would appreciate a description, or a reference.
On the wikipedia page they call the even covers the uniform covers, and mention that if we have an entourage-uniformity, the set of covers that are even w.r.t. to an entourage form the equivalent uniformity in the Tukey covering sense, and they also describe the inverse (going from covers to entourages).
I think (as to paracompactness) Kelley wanted an analogue between "paracompact iff every open cover has an open star refinement" and "paracompact iff every open cover has an even subcover" (wrt the uniformity of all open neighbourhoods of the diagonal, which is a uniformity when the space is paracompact (Hausdorff), see Kelley's exercise 6L, p 208 and moreover a paracompact space is complete in that uniformity), which to show a nice conceptual connection between uniform spaces theory and paracompactness properties.
This paper from 1978 does explore some generalisations of even covers and properties related to paracompactness.