Hello,

we have the following setting:

c:[0,1]->M a continuous path in a manifold M. I want to show that there has to be a finite open cover of coord. domains $\displaystyle (U_i)_{1,..,n}$ of c[0,1] , s.t. $\displaystyle U_i \cap c[0,1]=c(a_i,b_i) $ for ome $\displaystyle a_i, b_i.$

For sure, the image c[0,1] is compact in M. Therefore we can find a finite subcover to each open cover.

But why is the intercection of the finite cover with the curve of that form?

I think, we must first of all take open sets U_i, which has a connected intercection with c[0,1].

Why do we have such a cover?

I hope you can help me.

Regards