A semitopological group G is said to be an n-semitopological group, if for any \(g\in G\) with \(e\not \in \overline{\{g\}}\) there is a neighborhood W of e such that \(g\not \in W^{n}\) , where \(n\in \mathbb {N}\) . The class of n-semitopological groups ( \(n\ge 2\) ) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any \(n\in \mathbb {N}\) . Properties of n-semitopological groups are studied, and questions about n-semitopological groups are posed. Some generalized metric properties of n-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarser semi-metrizable topology; (2) each locally compact, Baire and \(\sigma \) -compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of n-semitopological groups are discussed.
