Let $(X,\tau)$ be a topological space. Then we know some conditions under which $(X,\tau)$ is metrizable (see for example this and this). It is also clear from these theorems that not every topological space is metrizable.
However, I am wondering whether the same is true for pseudometric spaces also. To be more specific,
Does there exist topological spaces which are not pseudometrazible?
Where we agree to call a topological space $(X,\tau)$ to be pseudometrizable iff there exists a pseudometric $d$ on $X$ such that the topology induced by the psuedometric is $\tau$.