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Fix $\epsilon>0$. I would like to understand how general are metric spaces in which open balls of radius $\epsilon$ are connected. One can give many examples assuming a vector space structure, but are there other useful examples?

A related question is whether in such a space balls of radius $\delta<\epsilon$ would also be connected?

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  • $\begingroup$ Path-metric spaces (in particular, connected Riemannian manifolds) provide many examples. $\endgroup$ – Moishe Kohan Jun 3 '17 at 16:33
  • $\begingroup$ The answer to your second question is no: take the unit circle (or a rescaling of it, to fit your $\epsilon$), and remove a single point from it. Then clearly there will be non-connected open balls of radius $\delta$ for sufficiently small $\delta$ $\endgroup$ – Max Jun 3 '17 at 19:30

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