Given two points in a manifold, can i find compact path-connected set that contains both

Question

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.

Answer 1

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 $K=\{\gamma(t)\,:\,t\in[0,1]\}$.

Answer 2

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 $a,b\in K.$