The first problem is proved in two steps.
First, If is a compact set and then there are two disjoint open sets such .
Second, each point in is not in .
If use part one to get disjoint open sets .
The compactness of gives a finite collection of each.
The intersection of the is open and contains .
Problem #2. Use the co-finite topology to show it is false.
He's suggesting that you take with the cofinite topology. The important thing being that the resulting top. space isn't first countable.