Definition:Topological Group

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

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

Then $ \left({G, \circ, \tau}\right)$ is a topological group if:


 * $(1): \quad \circ: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ 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 $\left({G, \circ}\right)$ be a group.

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

Let $\psi: \left({G, \tau}\right) \times \left({G, \tau}\right) \to \left({G, \tau}\right)$ be defined by
 * $\psi \left({ x, y }\right) = x \circ y^{-1}$.

Then $\left({G, \circ, \tau}\right)$ is a topological group 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.

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