Why are these varieties GL$(n,k)$ and T$(n,k)$ closed and connected?

Suppose that $k$ is an algebraically shut area. GL$(n,k)$ is the basic straight team. It can be taken into consideration as $\{x \in \mathbb{A^{n^2}}: det(x) \neq 0\}$. Plainly, this is an open part of $\mathbb{A^{n^2}}$. Yet why is it shut? and also why is it attached?

Allow T$(n,k)$ represents the subgroup of GL$(n,k)$ contains angled matrices. It is plainly shut. Yet just how to confirm its connectedness?

Several many thanks ~

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2022-06-07 14:33:00
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