Consider the two point set $X=\{a,b\}$ The possible topologies that can be found from X as follows. $$\begin{eqnarray} \tau_1&=&\{X,\emptyset\} &\text{Indiscrete topology} \\ \tau_2&=&\{X,\emptyset,{a}\} \\ \tau_3&=&\{X,\emptyset,{b}\} \\ \tau_4&=&\{X,\emptyset,{a},{b}\} &\text{Discrete topology} \end{eqnarray}$$
It is given that the trivial topology is Pseudometric and Discrete topology is a metric space.Also, $\tau_2$ and $\tau_3$ are known as Sierpinski space. Can you please explain me the above facts?
Further,I know that $\tau_2$ and $\tau_3$ are not $T_1$. But can they be Pseudo metric?