5
$\begingroup$

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.

$\endgroup$
8
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.