Just write down what open and closed sets are in this topology.
Let X be the infinite set. Fix .
Open sets A :
1.b) or is finite
Closed sets B :
2.a) (complement of a set in which p is not)
2.b) or is finite.
So now try to combine...
It's obvious that a set can't satisfy both 1.a) and 2.a), and can't satisfy both 1.b) and 2.b)
Now is it possible for a set to satisfy 1.a) and 2.b) ? Of course ! A finite subset of X which doesn't contain p will satisfy the two conditions. Thus it's both closed and open.
Similarly, it's possible to find a set that satisfies 1.b) and 2.a)...
A set that is neither open nor closed ? No it's not possible, because either p is in the set, either it's not (that's the law of excluded middle ). If it is, then the set is closed, and if it's not, then the set is open !