I would appreciate help clarifying:
On a metric space, connected does not imply path connected (e.g. topologist's sine curve). But I saw a theorem that if an open subset of the complex field is connected, then it is polygonally connected.
I can see that the topologist's sine curve is closed. What is it in general that allows an open set that is connected to imply path connected, and what prevents a connected closed set from being path connected.
Thanks from a self-studier