Finding a sequence of sets whose limit is a given set

Question

Suppose $B$ is a ball in $\mathbb{R}^{n}$ with $n>1$, and $F$ a closed set with empty interior. Is there an increasing sequence of sets $E_{1}\subset,...E_{n}\subset E_{n+1},...$, all Borel sets, such that each $E_{n}\subset B\setminus F$ but $\cup_{n}E_{n}=B$?

Answer 1

No.

In general if $A_\alpha$ are subsets of a set $D$, then $\bigcup_{\alpha} A_\alpha$ is also a subset of $D$, since any element of the union is an element of some $A_\alpha$ and hence an element of $D$.

Thus as Lord Shark the Unknown points out in the comments, if $B\cap F$ is nonempty then $B\setminus F$ is a proper subset of $B$, so since each $E_n$ is a subset of $B\setminus F$, we have that $\bigcup_n E_n \subseteq B\setminus F \subsetneq B$.