Special linear group is for $n\geq 1$ closed, but for $n=1$ also compact

Question

Let be

$$SL_n(\mathbb{R}):=\{A\in \mathbb{R}^{n\times n}|\det(A)=1\},\quad n\geq 1$$

a special linear group. Prove that $SL_n(\mathbb{R})$ is for $n\geq 1$ closed in $\mathbb{R}^{n\times n}\cong\mathbb{R}^{n^2}$ and for $n=1$ also compact.

Well, I know that $SL_n(\mathbb{R})$ is compact $\Longleftrightarrow SL_n(\mathbb{R})$ is bounded and closed.

Answer 1

It is the preimage of the closed set $\{1\}$ under the continuous map $A\mapsto \det A$. For $n=1$ you may try to enumerate all elements