# Interpolating arc in path-connected set of ${\mathbb R}^n$

I know that in ${\mathbb R}^n$ (as in any Hausdorff space), a path-connected subset $A$ is automatically arc-connected also.

Is it also true that, given any finite subset $B$ of $A$, there is an arc passing through all the points of $B$?

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No, consider a cross in $\mathbb{R}^2$ as $A$ and the set of its four endpoints as $B$.

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I slightly changed the word order in your sentence because, at first, I had some trouble parsing the sentence. If you dislike it, just change it back. +1 anyway –  Rasmus Dec 5 '12 at 9:30
It's better now, thank you! –  Thomas Dec 5 '12 at 10:12