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 $(x, y)\in U^n$?
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