# Definition:Topological Group

## Definition

Let $\left({G, \odot}\right)$ be a group.

On its underlying set $G$, let $\left({G, \tau}\right)$ be a topological space.

### Definition 1

$\left({G, \odot, \tau}\right)$ is a topological group if and only if:

$(1): \quad \odot: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ is a continuous mapping
$(2): \quad \phi: \left({G, \tau}\right) \to \left({G, \tau}\right)$ such that $\forall x \in G: \phi \left({x}\right) = x^{-1}$ is also a continuous mapping

where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.

### Definition 2

Let the mapping $\psi: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be defined as:

$\psi \left({x, y}\right) = x \odot y^{-1}$

$\left({G, \odot, \tau}\right)$ is a topological group if and only if:

$\psi$ is a continuous mapping

where $\left({G, \tau}\right) \times \left({G, \tau}\right)$ is considered as $G \times G$ with the product topology.

## Equivalence of Definitions

That the definitions presented above are equivalent is shown on Equivalence of Definitions of Topological Group.

## Also defined as

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

## Also see

• Results about topological groups can be found here.