# is the set of all $n×n$ matrices with trace zero is nowhere dense?

Is the set of all $n×n$ matrices with trace zero is nowhere dense?

I guess it is true as the image space is nowhere dense in $\mathbb{R}$. but can anybody confirm me please

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It's a hyperplane... – Martin Jan 19 '13 at 17:26
Nowhere dense in what? Surely the set is everywhere dense in itself in the usual topology. – user1551 Jan 19 '13 at 17:32

The set of matrices with trace zero is closed, because the trace is continuous. So you only have to show that the interior of your set is empty, and because the trace is linear, it is sufficient to find a sequence of matrices with nonzero trace convering to $0$.