Semitopological group

Remarks

Example: $(\Z, +)$ with the cofinite topology. Each translation $x \mapsto x + a$ is a bijection, so it maps cofinite sets to cofinite sets and is continuous. This group is not a topological group: every nonempty basic open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$, so the preimage of $\Z \setminus \set{0}$ under addition is not open. Non-example: the multiplicative group $\R \setminus \set{0}$ with the topology whose open sets are $\emptyset$, the whole set, and the sets $(a, \infty) \setminus \set{0}$ for $a \in \R$. The left translation $x \mapsto (-1) x$ is not continuous: the preimage of the open set $(1, \infty)$ is $(-\infty, -1)$, which is not open. Every topological group is a semitopological group, since joint continuity of multiplication implies separate continuity. In a semitopological group each translation is a homeomorphism, with inverse the translation by $a^{-1}$.