On the existence and properties of Alexandroff paratopological groups

We study groups endowed with Alexandroff topologies and show that no non-discrete Alexandroff topology can turn a group into a topological group. This settles negatively the basic existence problem for Alexandroff topological groups. Motivated by this obstruction, we turn to the broader setting of Alexandroff paratopological groups. We establish several fundamental properties of these spaces and provide explicit non-compact $T_0$ examples, showing that the Alexandroff framework is rich enough to capture nontrivial paratopological phenomena. As applications, we address two classical open questions concerning feebly bounded subsets in paratopological groups, proving that non-compact Alexandroff paratopological groups offer a positive solution both for products of feebly bounded sets and for the feebly boundedness of $B^2$ when $B$ is a feebly bounded subset.