In my third semester analysis course, our professor told us (without proof) that an open, connected subset $U\subset\mathbb{R}^n$ is automatically path connected. I understand the intuition behind the statement and I was able to prove it by defining for $x\in U$ the set $U(x):=\{y\in U\mid\exists\gamma\in C^0([0,1],X):\gamma(0)=x,\gamma(1)=y\}$, then showing that this set is open and thereby constructing a separation of $U$ under the assumption that it isn't path connected. I now asked myself wether there exist generalizations of this proposition. My proof heavily relied on the convexity of the open balls and thus on the vector-space structure of $\mathbb{R}^n$. Is it possible to generalize it, with some restrictive assumptions, to metric or even topological spaces? I'm aware of this question, but I consider the answer to it incomplete, so any further clarifications would be nice :)
Are there generalizations of "connectedness $\implies$ path-connectedness" in $\mathbb{R}^n$?
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general-topology
connectedness
path-connected
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0Yes I know, connectedness is in general weaker; this is why I asked for generalizations "with restrictive assumptions". – 2017-01-22
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1In every locally path-connected space, an open subset is connected if and only if it is path-connected. – 2017-01-22
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0For locally euclidean spaces connected$\implies$path connected, you can prove it essentially as you did for $\Bbb R^n$ – 2017-01-22
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0[A connected, locally path connected topological space is path connected](http://planetmath.org/sites/default/files/texpdf/39090.pdf). – 2017-01-23