  In this article topologies on metagroups are studied. They are related with
generalized $C^*$-algebras over ${\bf R}$ or ${\bf C}$. Homomorphisms and
quotient maps on them are investigated. Structure of topological metagroups is
scrutinized. In particular, topologies on smashed products and smashed twisted
wreath products of metagroups are scrutinized, which are making them
topological metagroups. Moreover, their inverse homomorphism systems are
studied.
