Definition:Topological Group/Definition 1

Definition
Let $\struct {G, \odot}$ be a group.

On its underlying set $G$, let $\struct {G, \tau}$ be a topological space.

$\struct {G, \odot, \tau}$ is a topological group :


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

where $\struct {G, \tau} \times \struct {G, \tau}$ is considered as $G \times G$ with the product topology.

That is, $\struct {G, \odot, \tau}$ is a topological semigroup with a continuous inverse.

Also see

 * Equivalence of Definitions of Topological Group