My analysis course states that a connected open set is path connected but I haven't been able to prove it. Here we consider a subset of the field of complex number but I suppose it works for a general topological space. Cheers
A connected open set is path connected
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$\begingroup$
general-topology
connectedness
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1It doesn't work for arbitrary topological spaces, but it works for locally path-connected spaces. – 2017-01-12
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0The proof is here: https://proofwiki.org/wiki/Connected_Open_Subset_of_Euclidean_Space_is_Path-Connected – 2017-01-12
1 Answers
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Hint: Show that if $U$ is an open set in a normed space, then for each $x \in U$ the set of points that are path-connected to $x$ is open and closed (relative to $U$).