Exercise of topological spaces

Question

$X$ is an infinite set and $T$ topology of $X$ in which all the infinite subset of $X$ are open, prove that $T$ is the discrete topology of $X$

Answer 1

Hints: 1) It suffices to show every singleton set $\{x\}$ is open. 2) To see that that is the case, can you exhibit $\{x\}$ as an intersection of two infinite subsets of $X$?

Answer 2

Let $A$ a set. We want to prove that $A$ is open. If $A$ is infinite you are done. Otherwise, $X-A$ is infinite and you can divide it in two infinite disjoint subsets $U_1$ and $U_2$. Then, $U_1\cup A$ and $U_1\cup A$ are infinite too, so

$$ A = (U_1\cup A)\cap (U_2 \cup A) $$ is open.