Abstract
Quasi-topological algebras, i.e., universal algebras with a topology with respect to which all operations are separately continuous, are studied. A construction of the free quasi-topological universal algebra \(F_\mathscr{V}(X)\) of an arbitrary Tychonoff space \(X\) in a given full variety \(\mathscr{V}\) of quasi-topological algebras is presented. It is proved that \(F_\mathscr{V}(X)\) has the inductive limit topology with respect to the natural decomposition into sets of polynomials of step \(\le n\) for \(n\in \omega\) . Questions related to the separation axioms satisfied by quasi-topological algebras are also considered. It is proved that every Tychonoff space \(X\) with a separately continuous Mal’tsev operation is homeomorphic to a retract of a Tychonoff quasi-topological group.
