Definition 9.1.1 (Topological Group).label Let $G$ be a group and $\mathcal{T}\subset 2^{G}$ be a topology. If

1. (TG1)

   The composition map $G \times G \to G$ with $(g, h) \mapsto gh$ is continuous.

2. (TG2)

   The inversion map $G \to G$ with $g \mapsto g^{-1}$ is continuous.

then the pair $(G, \mathcal{T})$ is a topological group.