Spivak proof partition of unity: case 2 (countable compact sets)

Question

I have a question about Spivak's proof on the partition of unity (see below). Spivak first proves the case where $A$ is compact. Then he proceeds with case 2: I don't really understand what this $U$ here is. Is $U$ simply an open subset of $\mathcal O$? And we consider all open subsets $U$ then? If so, what do we need this $U$? Why can't we just take $$ \mathcal O\cap(\operatorname{interior}A_{i+1}-A_{i-2}), $$ where $\mathcal O$ is an open cover of $A$.

Answer 1

$\mathcal O$ is an open cover, so by definition it is a family of open sets. That is $$\mathcal O = \lbrace U_\beta\rbrace_{\beta\in B},$$ where $B$ is just some set and $U_\beta$ are open. Then for each $i$ he creates a new family $$\mathcal O_i= \lbrace U_\beta \cap(\operatorname{interior}A_{i+1}-A_{i-2}) \rbrace _{\beta\in B}$$ and claims that this new family $\mathcal O_i$ is an open cover (but for other set).