Definition:Topological Group

Definition

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

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

Definition 1

$\struct {G, \odot, \tau}$ 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.

Definition 2

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 defined as

Some sources insist that a topological group be a Hausdorff space.

Also see

• Equivalence of Definitions of Topological Group

• Definition:Topological Semigroup
• Definition:Topological Ring
• Definition:Topological Field
• Definition:Topological Vector Space

• Results about topological groups can be found here.