It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean topology. What happens if we drop the Hausdorff condition, can we still classify all topological vector spaces? For example, there is one $\mathbb{C}^n_{triv}$ with the trivial (indiscrete) topology, and more generally we have $\mathbb{C}^k_{triv} \times \mathbb{C}^{n-k}$ for $0 \leq k \leq n$. Are these all?

Here is how one might start. If $V$ is a topological vector space of dimension $n$, then we can consider its subspace $K=\overline{\{0\}}$, which has some dimension $k$. The quotient $V/K$ is a topological vector space of dimension $n-k$. But it is also Hausdorff, so that it is isomorphic to $\mathbb{C}^{n-k}$. One checks that $K$ has the trivial topology (if $C \subseteq K$ is closed and $p \in C$, then $C-p \subseteq K$ is closed and contains $0$, so that $C-p=K$ and hence $C=K+p=K$.) But it is not clear a priori if this quotient splits.