A compact paratopological group is a topological group. How to prove it?
An abelian paratopological group is a topological group. Is this right?
A paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group $G$ with a topology such that the group's product operation is a continuous function from $G × G$ to $G$. This differs from the definition of a topological group in that the group inverse is not required to be continuous.
Thanks a lot.
The answer to the first question is yes, even if $G$ is not assumed Hausdorff. For a proof, see this answer. The $T_1$ case is elementary, though the general case seems to require some work.
The answer to the second question is no. Another answer in the same thread gives an example of a topology on $\mathbb{R}$ for which addition is continuous, but $x\mapsto -x$ is not continuous.
If you want an example of an abelian Hausdorff paratopological group that is not a topological group, there appears to be an example in the first link I posted, which uses some kind of $p$-adic positive cones on $\mathbb{Z}$.