# Uniform Structure on the Space of Subsets (Hyperspaces)

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.

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 $${\cal P}(X)$$.
• Oooh yeah, now I can see, thank you very much! This was simple indeed. So to put in the words of definition 1.6 should it be $$\{(E,F) \in 2^X \times 2^X \ | \ F \subset \mathcal{B}_\alpha (E) \ \text{and} \ E \subset \mathcal{B}_\alpha (F) \}?$$ Commented May 10, 2021 at 18:57
• Hello @HennoBrandsma thank you again for anwering me! Well, since you said that I'm trying to build a uniform covering from the definition, I set up $$\mathcal{B}_\alpha=\bigcup_{E \in 2^X} \mathcal{B}_\alpha (E)$$ and now I'm trying to prove that $$\mathcal{B}=\bigcup_{\alpha \in A} \mathcal{B}_\alpha$$ is the uniform covering because every $\mathcal{B}_\alpha$ will be a covering. Commented May 12, 2021 at 23:13