Example 3.3:

The set $GL(n, \mathbb{R})$ of all $n\times n$ invertible matrices with real entries is disconnected as a subspace of the space of all $n\times n$ matrices with real entries (the latter may be identified with $\mathbb{R}^{n^2}$).

If $GL(n, \mathbb{R})$ were connected then so would be its image under a continuous map. Well, the determinant map $d : GL(n, \mathbb{R}) \longrightarrow \mathbb{R}$ is continuous but the image is the real line minus the origin. The same argument shows that the set of all $n\times n$ orthogonal matrices $O(n, \mathbb{R})$ is disconnected.