# Examples of metric spaces with connected balls

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?

• Path-metric spaces (in particular, connected Riemannian manifolds) provide many examples. – Moishe Kohan Jun 3 '17 at 16:33
• 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 $\delta$ – Max Jun 3 '17 at 19:30