Regular and complete spaces over topological quasigroups

This article discusses the structure of topological spaces: We study regular and complete spaces over topological quasigroups and quasirings, as well as their embeddings and separations, homeomorphisms and isomorphisms, as well as relations between compactifications and the completeness of spaces over topological quasigroups. We also consider applications to topological spaces of families of mappings over topological quasigroups and quasirings together with their restrictions, openness, and continuous bijections. Some necessary theorems are established about the structure of topological quasigroups.