I'm studying the article "Topology on Spaces of Subsets" by Ernest Michael but I can't understand how he defines the uniform structure on on the space of non-empty closed subsets (a hyperspace) of a uniform space. The definition is in the image below.
OBS: $$V(x) = \{y \in X \ | \ (x,y) \in V\} $$ $$\langle V (x) \rangle_{x \in E} = \left\{F \in 2^x \ | \ F \subset \bigcup_{x \in E}V(x) \ \text{and} \ F \cap V(x) \ne \emptyset \ \forall \ x \in E \right\}$$
Here is what I got: First of all, this set $A$ indexing makes me more confused. I'm assuming that it is just a way to label the elements so we can call them if needed. I tried taking cartesian products between these $\langle V (x) \rangle_{x \in E}$ for all $V \in U$ and maybe use it as a base for the uniform topology but they should contain the diagonal so I got lost. To be honest, I'm new to uniform structures and searched a lot but nothing helped me. I found other articles and read a bit of General Topology by Stephen Willard bu even there I don't get the idea of how the set it describes can form a base, since I can't see how the diagonal is there for every element, here's the definition:
I really appreciate any help, be it explanation or indication of literature. I tried checking on Bourbaki but I couldn't find the 1940 version of the book which Ernest cites and the other versions I got aren't matching the citation.
Thanks in advance!
The idea is simple. Let $\cal U$ be an uniformity on a set $X$. For each $U \in {\cal U}$, let $$ \tilde{U} = \{(A, B) \in {\cal P}(X) \times {\cal P}(X) \mid A \subseteq U(B) \text{ and } B \subseteq U(A)\} $$ Note that since $A \subseteq U(A)$, $\tilde{U}$ contains the diagonal of ${\cal P}(X)$. Then the sets of the form $\tilde{U}$ form the base of a uniformity on ${\cal P}(X)$. It induces a uniformity on the set of nonempty closed subsets (resp. compact closed, finite subsets) of ${\cal P}(X)$.