So we have a manifold $M$ with nonempty boundary. Let $p,q\in\partial M$. I need to prove that I can connect $p$ and $q$ with a curve $\gamma:[a,b]\rightarrow M$ (such that $\gamma(a)=p$ and $\gamma(b)=q$) and for every $t\in(a,b)$ we have $\gamma (t)\notin\partial M$. I don't really know where to start with it. Any hints?
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https://math.stackexchange.com/questions/1145293/connected-manifolds-are-path-connected – Osama Ghani Jun 25 '20 at 00:22
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This can only be true if $M$ is connected. Do you consider topological or smooth manifolds? Do you know what a collar is? – Paul Frost Jun 26 '20 at 16:22