  The article is devoted to a structure of topological spaces related with
topological quasigroups. Regular and complete spaces over topological
quasigroups are studied. Separations and embeddings are also investigated for
them. Their homeomorphisms and isomorphisms, relations between
compactifications and completeness are scrutinized over topological
quasigroups. There are investigated applications to families of maps over
topological quasigroups, restrictions of mappings, their openness, continuous
bijections. Necessary theorems about topological quasigroups are proved.
