Definition:Topological Group/Definition 2

Definition

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

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

Let the mapping $\psi: \struct {G, \tau} \times \struct {G, \tau} \to \struct {G, \tau}$ be defined as:

$\map \psi {x, y} = x \odot y^{-1}$

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

$\psi$ is a continuous mapping

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

Also see

• Equivalence of Definitions of Topological Group

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