# Definition:Topological Semigroup

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## Definition

Let $\struct {S, \circ}$ be a semigroup.

On that same underlying set $S$, let $\struct {S, \tau}$ be a topological space.

Then $\struct {S, \circ, \tau}$ is a **topological semigroup** if and only if:

- $\circ: \struct {S, \tau} \times \struct {S, \tau} \to \struct {S, \tau}$ is a continuous mapping

where $\struct {S, \tau} \times \struct {S, \tau}$ is considered as $S \times S$ with the product topology.

## Also see

- Topological Group, an extension of this concept to a group.

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