Definition 5.1:

A topological group is a group which is also a topological space such that the singleton set containing the identity element is closed and the group operation

$$G\times G \longrightarrow G$$

$$(g_1, g_2)\mapsto g_1g_2$$

and the inversion $j: G \longrightarrow G$ given by $j(g) = g^{-1}$ are continuous, where $G\times G$ is given the product topology.

We leave it to the reader to prove that a topological group is a Hausdorff space. It is immediate that the following maps of a topological group $G$ are continuous:

1. Given $h\in G$ the maps $L_h: G\longrightarrow G$ and $R_h : G\longrightarrow G$ given by $L_h(g) = hg$ and $R_h(g) = gh$. These are the left and right translations by $h$.
2. The inner-automorphism given by $g \mapsto hgh^{-1}$ which is a homeomorphism.

Note that the determinant map is a continuous group homomorphism from $GL_n(\mathbb{R}) \longrightarrow R - \{0\}$. The image is surjective from which it follows that $GL_n(\mathbb{R})$ is disconnected as a topological space.