Some generalized metric properties of n-semitopological groups

A semitopological group G is called { an n-semitopological group}, if for any g∈ G with e∈\ there is a neighborhood W of e such that g∈ Wⁿ, where n. The class of n-semitopological groups (n≥ 2) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any n. Some properties of n-semitopological groups are studied, and some 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 Hausdorff semi-metrizable topology and has Gδ^-diagonal; (2) each locally compact, Baire and σ-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.