Hello,

Just write down what open and closed sets are in this topology.

Let X be the infinite set. Fix .

Open sets A :

1.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 !