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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 $[0,1]\to X$).

Alex Ravsky
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Mohamed Ali
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  • Path connected implies connected as all that can be show. – William Elliot Feb 15 '20 at 17:21
  • @WilliamElliot If I understand correctly you are saying that "Every path-connected space is connected" (in any topological space), thus any path-connected set is connected. Therefore the only interesting part of the claim in the Q is the other direction. – Mohamed Ali Feb 15 '20 at 17:25
  • Linked and linked. – Alex Ravsky Feb 15 '20 at 17:43
  • A classic fact is that in a locally path-connected space, an open subset of $X$ is path-connected iff it is connected. This generalises the $\Bbb R^n$ and other locally convex spaces situation, where we can replace path-connectedness even by polygonally connected (which is handy for integrals in complex analysis e.g.). – Henno Brandsma Feb 15 '20 at 22:43

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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.

Alex Ravsky
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