I need a concrete example to show this equality is false

Question

If $(X,\tau_{1})$ and $(X,\tau_{2})$ are $T_0$-space then $(X,\tau_{4})$ is not necessarily a $T_0$-space . Give a concrete example.

If $\tau_{4}. $is defined as $\tau_{4}$=$\tau_{1} \cap \tau_{2}$ then $\tau_{4}$ is a topology on $X$.

( That I did)

Answer 1

Take $X=\{0,1\}$, $\tau_1=\bigl\{\emptyset,\{0\},X\bigr\}$, and $\tau_2=\bigl\{\emptyset,\{1\},X\bigr\}$, in which case $\tau_1\cap\tau_2=\{\emptyset,X\}$.