My first question is
- Does every compact smooth manifold $M$ admit a finite atlas $\{(U_i,\varphi_i)\}_{i=1}^n$?
My attempt: Let $\{(U_\alpha,\varphi_{\alpha})\}_{\alpha \in I}$ be an atlas of $M$, then $\{U_\alpha\}_{\alpha\in I}$ is an open cover of $M$. Hence, it admits a finite subcover $\{U_{i}\}_{i=1}^n$ (slight abuse of notation here).
Does this sound right?
My second question is more about the number of size of an atlas of a surface:
- Let $M$ be a connected, bounded smooth surface embedded in $\mathbb{R}^3$. Is there are an upper bound on the smallest number of charts needed to cover $M$?
I found an article giving an upper bound of $2\cdot 9^2$ (or $9^2$ for surfaces without boundary). However, I am wondering if there are other results that give a tighter bound.