Proving that the set of $2\times 2$ orthogonal matrices is closed and disconnected as a subspace of all $2\times2$ matrices

Question

Let $O_2(\mathbb{R}) = \{A\in M_2(\mathbb{R})\mid AA^T=A^TA=I\}$ with the subspace topology induced by $M_2(\mathbb{R})$. Prove $O_2(\mathbb{R})$ is closed and disconnected. Then conclude from it that there is no continuous curve like $\rho:[0,1] \rightarrow O_2(\mathbb{R})$ such that $\rho(0)=I$ and $\rho(1)=\left( \begin{array}{ccc} 0 &-1 \\ -1&0\end{array}\right)$ .

Answer 1

The inverse image of a closed set (e.g. the set containing only $I$) under a continuous function is closed.

As for "disconnected", do you know how to prove that the determinant of an orthogonal matrix is either $1$ or $-1$?