The first problem is proved in two steps.
First, If $\displaystyle B$ is a compact set and $\displaystyle p\notin B$ then there are two disjoint open sets such $\displaystyle B\subset G~\&~p\in H$.
Second, each point in $\displaystyle A$ is not in $\displaystyle B$.
If $\displaystyle a\in A$ use part one to get disjoint open sets $\displaystyle G_a~\&~H_a$.
The compactness of $\displaystyle A$ gives a finite collection of each.
The intersection of the $\displaystyle G_x's $ is open and contains $\displaystyle B$.
Problem #2. Use the co-finite topology to show it is false.
He's suggesting that you take $\displaystyle \mathbb{R}$ with the cofinite topology. The important thing being that the resulting top. space isn't first countable.