Topological spaces which are not pseudometrizable.

A pseudometric space is symmetric (also called $R_0$): if $x \in \overline{\{y\}}$ then $y \in \overline{\{x\}}$ (basically because $d(x,y)=0$ implies $d(y,x)=0$ too, also in pseudometric spaces).

Sierpinski space ($X=\{0,1\}$ with topology $\{\emptyset,\{0\},X\}$) is not symmetric so not pseudometrisable. (Because $1 \in \overline{\{0\}}$ but not the other way around). This is in a way the simplest example, certainly the smallest one.

If $X$ is $T_1$ then $X$ is metrisable iff $X$ is pseudometrisable. (the $T_1$ ensures that $X$ is also $R_0$ and so the non-existence of any points $x,y$ with $x \neq y$ but $d(x,y)=0$. So the pseudometric for $X$ on the right is then a metric.)

So spaces like the cofinite topology on $\mathbb{N}$ is not pseudometrisable, as it's not metrisable (not Hausdorff to start with...) Also, the Sorgenfrey line, the Michael line, double arrow space etc etc.

The Sierpinski space is not pseudometrizable. The Kolmogorov quotient of a pseudometric space is metric. However the Sierpinski space is already $T_0$, but it's not Hausdorff, and thus not metric.

The Sierpinski space is the topological space on two points with the topology $$\{\varnothing, \{0\}, \{0,1\}\}.$$