Suppose that $k$ is an algebraically closed field. GL$(n,k)$ is the general linear group. It can be considered as $\{x \in \mathbb{A^{n^2}}: det(x) \neq 0\}$. Clearly, this is an open subset of $\mathbb{A^{n^2}}$. But why is it closed? and why is it connected?
Let T$(n,k)$ denotes the subgroup of GL$(n,k)$ consists of diagonal matrices. It is clearly closed. But how to prove its connectedness?
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