Topological semigroup

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.

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 $g\circ 0$ is constantly zero. Thus right-translation continuity alone does not imply continuity of the other translations.

A left-topological semigroup is a semigroup with a topology for which every left translation $L_a:x\mapsto ax$ is a continuous map. Here $a$ is fixed and the right input varies. Requiring continuous right translations $R_a:x\mapsto xa$ instead gives a right-topological semigroup.^[ref]

Every nonempty compact Hausdorff space carrying a left-topological semigroup structure contains an idempotent element of a semigroup.

Choose an inclusion-minimal nonempty compact subsemigroup $K$, using Zorn's lemma and the finite intersection property. For $a\in K$, the set $aK$ is a nonempty compact subsemigroup, so minimality gives $aK=K$. Hence $F=\{x\in K:ax=a\}$ is nonempty. It is a compact closed subset of $K$ by continuity and the Hausdorff space assumption, and is a subsemigroup because $a(xy)=(ax)y=ay=a$ for $x,y\in F$. Minimality gives $F=K$, so $a^2=a$.^[ref]

Reversing multiplication gives the same conclusion for a right-topological semigroup. Joint continuity of multiplication is not needed.