# Definition:Topological Group

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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.

### 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

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

- Results about
**topological groups**can be found**here**.