The goal is to prove that if a tvs $(X,\tau)$ is Haussdoff and of dimension $n$, then it is homeomorphic to $\mathbb{R}^n$.
My attempt is to use the isomorphism $\phi$ (known in linear algebra) through the unique expression in terms of the base $$\begin{align}\phi:&X&\longrightarrow&\mathbb{R}^n\\&\sum_{j=1}^n\lambda_jx_j&\longrightarrow&\sum_{j=1}^n\lambda_je_j\end{align}$$ , with $\lbrace e_j\rbrace$ the canonical base of $\mathbb{R}^n$ and $\lbrace x_j\rbrace$ a base of $X$, so the problem is to prove that $\phi$ is bi-continuous. In the dirction $\mathbb{R}^n\longrightarrow X$ it's simply the composition of projection (which is continuous) and scalar product, that is continuous by definition. Until here I'm OK.
In the direction $X\longrightarrow\mathbb{R}^n$ I'm stack (since I can't simply "project" and don't have norm!). I've thought about proving continuity in $0_X$, which is $\forall r>0,\exists U\in N_{o_X}:\phi(U)\subset B(0,r)$ for some metric of $\mathbb{R}^n$, but I only can write the definition. The other attempt is to use some property of topological spaces, but couldn't find something close to open application in Banach spaces.
I would appreciate any hint or help. Thank you in advance.