Definition:Topological Group

Definition

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

On that same 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.

Property $(1)$ ensures that any topological group is also a topological semigroup.