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 $\leqslant n$ 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.
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