Topological group

A topological group is a group $G$ equipped with a topology such that:

1. The multiplication map $\mu: G \times G \to G$, $(g, h) \mapsto gh$, is continuous .
2. The inversion map $\iota: G \to G$, $g \mapsto g^{-1}$, is continuous.

Equivalently, the single map $(g, h) \mapsto gh^{-1}$ is continuous.

Properties

• Translation maps $L_g: h \mapsto gh$ and $R_g: h \mapsto hg$ are homeomorphisms .
• The topology is determined by neighborhoods of the identity.
• Connected components form a normal subgroup .

Examples

• $(\mathbb{R}, +)$ and $(\mathbb{R}^n, +)$ with the Euclidean topology.
• $(\mathbb{R}^*, \cdot)$ (nonzero reals under multiplication).
• $GL_n(\mathbb{R})$ and $GL_n(\mathbb{C})$ with the subspace topology.
• Lie groups are smooth topological groups.
• Any group with the discrete topology.

Relation to Lie groups

A Lie group is a topological group that is also a smooth manifold with smooth group operations.