Let $Dom$ be a uniform space, and $f$ be a continuous function from $Dom$ to itself satisfying:

 1. For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural number $n$ and a member $x$ of $U$ such that $f^n(x)$ is a member of $V$.

 2. The periodic points of $f$ are dense in $Dom$.



Does it follow that $f$ satisfies (3)?

 (3)  There exists an entourage $U$ of $Dom$ such that for all members $x$ of $Dom$ and all neighborhoods $N$ of $x$, there exists a member $y$ of $N$ and a natural number $n$ such that $(f^n(x),f^n(y))$ is not a member of $U$.



According to [on_intervals](http://pb.math.univ.gda.pl/chaos/pdf/on_intervals.pdf), the implication holds in metric spaces.