Definition:Topological Group/Definition 1

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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:

\((1)\)   $:$   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.


Also see


Sources