Checking whether a given space is indiscrete or not

Question

Problem

Let X be an infinite set and τ a topology on X with the property that the only infinite subset of X which is open is X itself. Is (X,τ) necessarily an indiscrete space?

So I have to prove that set contains only two elements namely X and $\emptyset$ . It can be shown that sets like X-{x} are not in topology.

How to proceed?

Answer 1

The statement is false. Take, for example, $X=\mathbb{Z}$ and $\tau=\{\emptyset,\{0\},\mathbb{Z}\}$.

Answer 2

Hint: It's not true, look for a topology with one nontrivial (infinite) open set.