I was recently wondering whether there are conraction principle like theorems for uniform spaces but which are not metric spaces. The theorem relies heavily on the notion of completeness, which exists for non-metric uniform spaces.
My question to which I wasn't able to find a satisfactory answer from my current search, is whether there is such a theorem in the uniform but non-metric setting? Something like:
Let $f:X\to X$ be a uniformly continuous map on a complete uniform space $X$, such that $\big(f\times f\big)^{n}[E]\subseteq E'$ for any entourages $E,E'$ and $n>N(E,E')$. Then $f$ has a unique fixed point.
It seems natural to me given the setting of uniform spaces, which I am admittedly new to. Is there perhaps a counter example to such a theorem?