© 1996, Topology Atlas

One of the important questions of topological algebra is a question about how algebraic operations in this system influence the existence of distinct topologies, in particular, the question about a possibility of defining non-discrete topologies in the infinite algebraic systems? The beginning of this problem for countable groups research was set by A.A.Marcov in 1946. Further, this question was studied by many authors for different algebraic systems:

• For groups and their generalizations (Kertesz A., Szele T., Ol'shansky A., Hanson J., Suvorov N., Taimanov A., and others);
• For rings and fields (Arnaautov V., Hochster M., Kiltintn J., Mutylin A. and others). Proof of the main rezults on this question can be foud, for example, in the monograph of V.I.Arnautov, S.T.Glavatsky, A.V.Mikhalev "Introduction to the theory of topological Ring and Modules", Marcel Dekker, Inc. New York,Basil, Hong Kong 1996;
• Besides, great number of works are devoted to the questin aboud the number of different topologies on the algebraic system given.

State of the art at present:

• Each abelian group admits a non-discrete group topology;
• It was created an example of countable group, which does not admit non-discrete group topologies;
• Each countable (not necessarily associative ) ring admits a non-discrete ring topology;
• Each infinite associative-commutative ring admits non-discrete ring topology;
• It was created an example of non-associative infinite ring, which does not admit a non-discrete ring topologies;
• Each ifinite field admits non-discrete topology in which it is a topological field.

Problems:

• Does any infinite assotiative ring admit a non-discrete ring topology?
• Does any infinite (countable) skew field admit non-discrete topology in which it is a topological skew field?