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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.

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    How about the determinant map again, this time restricted to $C$?2012-03-07
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    Can you characterize the set of invertible upper triangular matrices inside the set of all upper triangular matrices?2012-03-07
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    The invertible upper triangular matrices are those which have nonzero determinant in $C$. Then apply the same argument as before and hence we have that $B$ is an open set?2012-03-07
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    Open in where, $C$ or $M(n)$?2012-03-07

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