Where is a non-empty open set connected iff path-connected? I would to know how can the following claim can be generalized:

Theorem. A nonempty open set in the plane is connected if and only
  if any two of its points can be joined by a polygon which lies in the set.

An example of generalisation would be plane$=\mathbb{R}^2$ into $\mathbb{R}^n$ into metric space (or some special type of metric space).
"joined by polygon", I assume, could be generalised to "path-connected" (meaning a continuous map $[0,1]\to X$).

According to Exersice 6.3.10 from “General Topology” by Ryszard Engelking (2nd ed., Heldermann, Berlin, 1989), a topological space $X$ is locally pathwise connected if for every $x\in  X$ and any neighbourhood $U$ of the point $x$ there exists a neighbourhood $V$ of $x$ such that for any $y\in V$ there exists a continuous mapping $f: [0,1]\to U$ satisfying $f(0)=x$ and $f(1)=y$. In the exercise is proposed the following. Verify that every locally pathwise connected space is locally connected and give an example of a locally connected subspace of the plane which is not locally pathwise connected (cf. Problem 6.3.11). Show that every connected and locally pathwise connected space [and so every open connected subspace of a locally pathwise connected space AR.] is pathwise connected.
