# Definition:Topological Semigroup

## Definition

Let $\left({S, \circ}\right)$ be a semigroup.

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

Then $\left({S, \circ, \tau}\right)$ is said to be a topological semigroup if:

$\circ: \left({S, \tau}\right) \times \left({S, \tau}\right) \to \left({S, \tau}\right)$ is a continuous mapping

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