= Right-topological semigroup
{title2=$R_a(x)=xa$}
A right-topological semigroup is a <semigroup> with a <topology> such that, for each fixed $a$, the right translation $R_a:x\mapsto xa$ is a <continuous map>. The varying input is on the left. Requiring $x\mapsto ax$ instead gives a <left-topological semigroup>; requiring multiplication jointly continuous gives a <topological semigroup>.
For a one-sided example, take all <functions> $\mathbb R\to\mathbb R$ with <function composition> and the <product topology>, so convergence is <pointwise convergence>. For fixed $g$, the map $f\mapsto f\circ g$ is continuous because its value at $x$ is the evaluation of $f$ at the fixed point $g(x)$.
Left composition can fail to be continuous. Let $g(x)=1$ for $x>0$ and zero otherwise, and let $f_n$ be constant $1/n$. Then $f_n$ converges pointwise to the zero <function>, but $g\circ f_n$ is constantly one while $g\circ0$ is constantly zero. Thus right-translation continuity alone does not imply continuity of the other translations.
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