Use of a Covering theorem

Question

where

I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified. The books gives no justification.

Answer 1

The set of closed balls$$\mathcal{B} = \{B[x,r] \colon x \in A, \,10r \leq \delta \text{ and } B(x,r) \subseteq (\mathbb{R}^n - \cup_{i = 1}^k B_i) \}$$ is a cover of $$A - \bigcup_{i = 1}^k B_i \, \subseteq \, \mathbb{R}^n - \bigcup_{i = 1}^k B_i $$ Why? Because if $x \in A - \bigcup_{i = 1}^k B_i$, this means that $x \in \mathbb{R}^n - \bigcup_{i = 1}^k B_i$, which is open, so there is some $q$ such that $20q \leq \delta$ and the open ball $B(x,q) \subset \mathbb{R}^n - \bigcup_{i = 1}^k B_i$, so effectively, shrinking this ball a bit ($r= q/2$) we can get a closed ball $B[x,r]$ inside the open ball, centered at $x$ which is obviously inside of $\mathbb{R}^n - \bigcup_{i = 1}^k B_i$, and this last closed ball, belongs to $\mathcal{B}$.

Then because the $\sup$ of radiuses in $\mathcal{B}$ is less tan a fixed number $\delta$, you can apply theorem $2.1$ (Vitali's) and find the desired countable many balls which makes the highlighted part true.