= Topological semigroup
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A <topological semigroup> is a <semigroup> equipped with a <topology> for which multiplication $S\times S\to S$ is a <continuous map>, using the <product topology> on $S\times S$. Thus the two inputs vary together. The <real numbers> under <addition> are an example.
Continuity of each translation separately is weaker. A <left-topological semigroup> requires continuous maps $x\mapsto ax$ for fixed $a$; a <right-topological semigroup> instead requires continuous maps $x\mapsto xa$. Stating which factor is fixed avoids confusing the two conventions.
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