Definition:Topological Group/Definition 1

Definition
Let $\left({G, \circ}\right)$ be a group.

On its underlying set $G$, let $\left({G, \tau}\right)$ be a topological space.

Then $\left({G, \circ, \tau}\right)$ is a topological group if:


 * $(1): \quad \circ: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ is a continuous mapping
 * $(2): \quad \phi: \left({G, \tau}\right) \to \left({G, \tau}\right)$ such that $\forall x \in G: \phi \left({x}\right) = x^{-1}$ is also a continuous mapping

where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.

That is, $\left({G, \circ, \tau}\right)$ is a topological semigroup with a continuous inverse.