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.

  • $\begingroup$ What you call c-coarsely connected is usually called c-chainable. $\endgroup$ Jul 7 '17 at 15:05
  • $\begingroup$ It is almost the same as "the space has connected completion". $\endgroup$ Jul 7 '17 at 15:46
  • $\begingroup$ 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? $\endgroup$ Jul 7 '17 at 16:03
  • 2
    $\begingroup$ Such a space is called chainable. $\endgroup$
    – Misha
    Jul 8 '17 at 12:20
  • $\begingroup$ Thank you for the comments. Misha, thanks for the answer! $\endgroup$ Jul 9 '17 at 14:23

While the term chainable is sometimes used for such spaces, it clashes with the established usage of "chainable" in continuum theory (e.g., Each homogeneous nondegenerate chainable continuum is a pseudo-arc by R. H. Bing).

The term chain connected is used in papers on discrete homotopies (e.g., Essential circles and Gromov-Hausdorff convergence of covers by Plaut and Wilkins) and in a few textbooks such as Introduction to General Topology by Joshi:

Incidentally, item (b) demonstrates the difference between being chain connected and having connected completion.


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