A metric space $(X,d)$ is called $c$-*coarsely connected* if for every two points $x,y\in X$ there exists a sequence $x=x_0,x_1,\ldots,x_n=y$ of points in $X$ such that $d(x_{i-1},x_i)\leq c$.

**Question:**
Is there an established terminology for spaces which are $c$-coarsely connected for every $c>0$?

**Examples:** connected spaces, $\mathbf Q$ with the standard metric induced from $\mathbf R$, $\mathbf Z$ with the metric induced from the circle via an injective homomorphism.

I am thinking of calling it *finely connected*.

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