Is there a $D$- chain between two point of a connected component in an uniform space?

A uniform space $$(X, U)$$ is said to be uniformly connected if every uniformly continuous map of the space into a discrete space is a constant map also a topological space is connected if and only if the only mapping of the space into a discrete space is a constant mapping. This implies that if uniform space $$(X, \mathcal{U})$$ is connected, then for each pair $$x,y\in X$$ and each $$U\in\mathcal{U}$$, there is an integer $$n$$ such that $$(x, y)\in U^n$$.

Let $$A\subseteq X$$ is a connected component ($$X$$ is not necessary connected). Let $$x,y\in A$$ and $$D\in\mathcal{U}$$ be given. Is there an integer $$n$$ such that $$(x, y)\in U^n$$?

• @HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $n\in\mathbb{N}$) of $\mathbb{Z}\times \mathbb{Z}$, where $(x,y)\in D_n$ if and only if $x\equiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it. – user479859 Dec 8 '18 at 20:45