Exercise: A topological space is said to be $T_0$-space if for each pair of distinct points $a,b$ in $X$, either there exists an open set containing $a$ and not $b$, or there exists an open set containing $b$ and not $a$.A topological space $(X,\tau)$ is said to be $T_1$-space if every singleton set $\{x\}$ is closed in $(X,\tau)$
Put a topology on the set $X=\{0,1\}$ so that $(X,\tau)$ will be a $T_0$-space but not a $T_1$-space.
If I consider the topology $\tau=\{X,\emptyset,\{0\}\}$, the point $0\in\{0\}$ and $\{0\}\in X$. If I come up with other topology since I need to have $X$ in the topology there will be always two open sets containg the same point.
Question:
So how do I build a topology on this case that $(X,\tau)$ will be a $T_0$-space but not a $T_1$-space? What am I thinking wrong?
Thanks in advance!