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:

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

  • Results about topological groups can be found here.