Let Dom be a uniform space, and f be a continuous function from Dom to itself satisfying:
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.
The periodic points of f are dense in Dom.
Does it follow that f satisfies (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 pb.math.univ.gda.pl/chaos/pdf/on_intervals.pdf, the implication holds in metric spaces.