Paratopological group

Remarks

Example: $(\R, +)$ with the topology of the Sorgenfrey line, generated by the intervals $[a, b)$. If $x + y \in [c, d)$, choose $\varepsilon > 0$ with $x + y + 2\varepsilon < d$; then $[x, x + \varepsilon) + [y, y + \varepsilon) \subseteq [x + y, x + y + 2\varepsilon) \subseteq [c, d)$, so addition is continuous. Inversion is not continuous, since the preimage of $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open. So this is a paratopological group that is not a topological group. Non-example: $(\Z, +)$ with the cofinite topology is a group, but addition is not jointly continuous, because every nonempty open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$. A paratopological group is the same as a topological monoid whose underlying monoid is a group.