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 if and only if the following conditions are fulfilled:

$(1)$	$:$		Continuous Group Product					$\odot: \struct {G, \tau} \times \struct {G, \tau} \to \struct {G, \tau}$ is a continuous mapping
$(2)$	$:$		Continuous Inversion Mapping					$\iota: \struct {G, \tau} \to \struct {G, \tau}$ such that $\forall x \in G: \map \iota x = x^{-1}$ 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 inversion mapping.