# $c$-coarsely connected space for every $c>0$

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.

• What you call c-coarsely connected is usually called c-chainable. Jul 7 '17 at 15:05
• It is almost the same as "the space has connected completion". Jul 7 '17 at 15:46
• It seems that existence of $c$-chain for any $c>0$ defines a coarse structure (en.wikipedia.org/wiki/Coarse_structure), why not to use it? Jul 7 '17 at 16:03
• Such a space is called chainable. Jul 8 '17 at 12:20
• Thank you for the comments. Misha, thanks for the answer! Jul 9 '17 at 14:23 