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

Please help me to know it.

Thanks

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    $\begingroup$ Yes, if you mean uniform-connected component. $\endgroup$ – Henno Brandsma Dec 8 '18 at 18:47
  • $\begingroup$ @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. $\endgroup$ – user479859 Dec 8 '18 at 20:45

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