Topological proof on discrete topology where $X$ is infinite
HINT: Since $X$ is infinite, there is an infinite $A\subseteq X$ such that $X\setminus A$ is also infinite. For convenience let $B=X\setminus A$. For each $x\in A$ consider the open sets $A$ and $\{x\}\cup B$. Do something very similar to handle the points of $B$.