Topological semigroup

Remarks

Example: $[0, 1]$ with ordinary multiplication and the topology from $\R$. The product is associative, it maps $[0,1] \times [0,1]$ into $[0,1]$, and it is continuous as the restriction of a polynomial. Non-example: the semigroup $(\Z, +)$ with the cofinite topology. Addition is associative, but it is not jointly continuous: the preimage of $\Z \setminus \set{0}$ is not open, since every nonempty basic open set $U \times V$ contains a pair $(x, -x)$. If one asks only that each translation $y \mapsto xy$ and $y \mapsto yx$ be continuous, the weaker structure is called a semitopological semigroup.