Definition:Topological Group

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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.