Olga V Sipacheva

Universal algebras with topology.

A universal algebra, or simply an algebra, is a nonempty set together with arbitrarily many operations (of finite arity). A topological (quasi-topological) algebra is an algebra endowed with a topology with respect to which all operations are continuous (respectively, separately continuous). Classes of (topological, quasi-topological) algebras closed with respect to the formation of subalgebras, products, and quotients are called varieties. A variety of all topological (quasi-topological) algebras whose underlying abstract algebras (that is, the same algebras without topology) belong to a given variety of abstract algebras is said to be full. In any variety of topological or quasi-topological algebras, the free topological algebra of any nonempty topological space is uniquely defined.

By a topological quotient of an algebra A with a topology we mean a homomorphic image of A endowed with the quotient topology. It is proved that any topological quotient of a quasi-topological algebra is a quasi-topological algebra, i.e., all operations are separately continuous with respect to the quotient topology. An example showing that this is not so in the case of topological algebras is given. Sufficient conditions on a variety of topological algebras under which any topological quotient of any algebra in this variety is a topological algebra are presented.

It is shown that properties of free quasi-topological algebras are nicer than those of free topological algebras in many respects. In particular, the free quasi-topological algebras F(X) of spaces X in full varieties of quasi-topological have the following features distinguishing them from free topological algebras:

  1. the quasi-topological algebra F(X) of any space X is the inductive limit of its subspaces F_n(X) of words of length at most n (defined in a standard natural way);
  2. if the number of operations is finite and X is Tychonoff, then the subalgebra of F(X) generated by any closed subspace Y of X is topologically isomorphic to F(Y);
  3. the quasi-topological algebra F(X) of any space X is a topological quotient of the absolutely free quasi-topological algebra (that is, the algebra of words) W(X) over the signature of the variety under consideration.

Embeddings of topological spaces in their free topological and quasi-topological algebras, as well as separation axioms in such algebras, are also considered.