$\tau_2=\{\emptyset,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}$
Let me think...
$\{b\}^C=\{a,c\}\not\in \tau_2$, thus $\{b\}$ is not closed
so there is some sequence in $\{b\}$ converging to a or c, which is not in $\{b\}$.
$\{a,b\}^C=\{c\}\not\in \tau_2$, thus $\{a,b\}$ is not closed
so there is some sequence in $\{a,b\}$ converging to c, which is not in $\{a,b\}$.
$\{a,b,c\}^C=\{a,b,c\}\in \tau_2$, thus $\{a,b,c\}$ is closed
so there is some sequence in $\{a,b,c\}$ converging to a or b or c, which are in $\{a,b,c\}$