Abstract
In this article, topologies on metagroups and quasigroups are studied. Topologies on smashed twisted wreath products of metagroups are scrutinized, which are making them topological metagroups. For this purpose, transversal sets are studied. As a tool for this, semi-direct products of topological metagroups are also investigated. They have specific features in comparison with topological groups because of the nonassociativity, in general, of metagroups. A related structure of topological metagroups is investigated. Particularly, their compact subloops and submetagroups are studied. Isomorphisms of topological unital quasigroups (i.e., loops) obtained by the smashed twisted wreath products are investigated. Examples are provided.
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