Olga Sipacheva

Quasi-Topological Universal Algebras‎‎.

Recall that an n-ary operation on a set A is any map A^n\to\nobreak A. A universal algebra is a nonempty set A together with a collection of operations. In what follows, by algebras we mean universal algebras.

The theory of topological algebras, that is, algebras with a topology with respect to which all operations are continuous, originates from the work of Mal’tsev and has been greatly developed. The talk is concerned with quasi-topological algebras, i.e., topological spaces on which separately continuous operations of various arities are defined. They are interesting for a number of reasons. First, quasi-topological algebras have good categorical properties, which are not inherent in topological algebras; for example, given any continuous map of a quasi-topological algebra to a topological space, there exists a minimal (in the natural sense) factorization of this map through a continuous homomorphism to another quasi-topological algebra of the same signature. Secondly, continuous operations on a topological space very rarely extend to continuous operations on the Stone–Čech compactification of this space; they extend to separately continuous operations much more often. Yet another fundamental difference between quasi-topological and topological algebras is that any topological quotient of a quasi-topological algebra is a quasi-topological algebra, that is, quotient homomorphisms preserve the continuity of operations. Just as in the case of topological algebras, in every variety of quasi-topological algebras, the free quasi-topological algebra of any topological space X is defined, but the topology of free quasi-topological algebras have a much more constructive description than that of free topological algebras. Moreover, many results concerning topological algebras remain valid and even become stronger for quasi-topological algebras. For example, any quasi-Mal’tsev space is a retract of a quasi-topological group, while a Mal’tsev space is a retract of a topological group only under additional conditions.

We discuss these and some other properties of quasi-topological algebras. One of our main results is that, given any T_1-space X and any variety \mathscr V of quasi-topological algebras, the subalgebra F_{\mathscr V}(Y|X) of the free quasi-topological algebra F_{\mathscr V}(X) on X generated by a closed subspace Y of X is the free quasi-topological algebra F_{\mathscr V}(Y) on Y. We also discuss necessary conditions on Y under which F_{\mathscr V}(Y|X) = F_{\mathscr V}(Y). One of such conditions is that Y is d-closed in X, that is, any disjoint closed subsets of Y have disjoint closures in X.