Let $E$ be a one-dimensional real Hausdorff TVS. We fix $v \in E \setminus \{0_E\}$ and consider the linear map $$ T: \mathbb R \to E, t \mapsto tv. $$
To prove that any $n$-dimensional Hausdorff TVS is homeomorphic to $\mathbb R^n$, I would like to verify
Theorem $T$ is a homeomorphism.
There are possibly subtle mistakes that I could not recognize in below attempt. Could you please have a check on it?
It's clear that $T$ is bijective continuous. It remains to prove that its inverse $$ T^{-1} : E \to \mathbb R, tv \mapsto t $$ is continuous. We need the following result.
Lemma Let $E$ be a (not necessarily Hausdorff) real TVS and $f:E \to \mathbb R$ linear. Then $\ker f$ is closed if and only if $f$ is continuous if and only if $f$ is continuous at $0$.
By above Lemma, it suffices to prove that $\ker T^{-1} = \{0_E\}$ is closed. This is true because $E$ is Hausdorff. This completes the proof.
