I want to know the counter example for: Hyperspace of an arbitrary complete uniform space need not be complete.
The hyperspace of a uniform space $(X,\mathscr D)$ is obtained by forming the set $\mathscr H$ of all non empty closed subsets of $X$ and taking as a base for a diagonal uniformity on $\mathscr H$, the collection of all sets of the form $\{(A,B): A$ is $D$-close to $B\}$, for $D$ in $\mathscr D$, where $A$ and $B$ are $D$-close iff $A\subset D[B]$ and $B\subset D[A]$.
I know that it is not true for metric spaces so I was trying an example of a uniform space which is not either separating or second countable. But I think for me uniform spaces are more difficult to deal with, especially the counter examples. It is an exercise 39D in the book "General topology",by Stephen Willard