I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions:

1- $(X,\tau)$ is not locally convex.

2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subsets of $X$ such that $X=\cup X_n$ and the topology $\tau$ on $X_n$ is relatively second countable and $d$-metrizable for every $n$.