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Suppose we are given two points in path-connected smooth manifold. My hypothesis is that we can find path-connected compact set that contains both. I have no idea how to prove it, in fact I don't know if it's true. Any hints?

I guess this question was really stupid, now that I know the answer :-) I don't know why I didn't see it immediately.

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  • $\begingroup$ Do you understand what path-connected means? $\endgroup$ – Peter Franek Aug 8 '15 at 16:22
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Say the two points are $p$ and $q$. By definition of "path-connected" there exists a continuous function $\gamma$ from $[0,1]$ to your manifold such that $\gamma(0)=p$ and $\gamma(1)=q$. Let $K=\{\gamma(t)\,:\,t\in[0,1]\}$.

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If $a,b\in M$ then there exists a continuous path $j:[0,1]\to M$ such that $j(0)=a, j(1)=b$ the set $K=j([0,1] )$ is compact and $a,b\in K.$

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