Let $B$ denote the $n \times n$ invertible upper triangular matrices. I am trying to duplicate the work done here where I asked a similar question for $GL_{n}(\mathbb{R})$.
My thought is: Let $C$ be the space of $n \times n$ upper triangular matrices, then $C \cong \mathbb{R}^{n(n + 1)/2}$. If $B$ is an open subset of $C$, then I am done, by the similar reasoning as in the $GL_{n}(\mathbb{R})$ case. However, I can't seem to think of a continuous map and a set that would give my such a result.