This question arise when i read the proof of Whitney Approximation Theorem for Function in Lee's book.
As i noticed, in the usual partition of unity arguments, e.g. extending smooth functions over a closed subset of a manifold, existence of Riemannian metric, etc., we only need a partition of unity of an uncountable open cover of the manifold. However, in the proof of Whitney Approximation Theorem (Lee's ISM), he explicitly use partition of unity over a countable cover.
I've looked at the details for a while now, but i can't find any reason why we should use the countable one. Did anyone know why ?
Any help will be appreciated. Thank you.
This appears to just be an oversight. The choice of the countable subcover $\{U_{x_i}\}$ is totally unnecessary, and you can just take a partition of unity subordinate to the open cover $\{U_0\}\cup\{U_x\}_{x\in M\setminus A}$.