Chapter 66 - Moscow questions on topological algebra

• Published in: Open Problems in Topology II pp. 711 - 726
• Main Authors: Kozlov, Konstantin L., Reznichenko, Evgenii A., Sipacheva, Ol′ga V.
• Format: Book Chapter
• Language: English
• Published: Elsevier B.V 2007
• ISBN: 9780444522085; 0444562745; 0444522085; 9780444562746; 0080475299; 9780080475295
• DOI: 10.1016/B978-044452208-5/50066-1

This chapter discusses some questions on topological algebra. The chapter begins with the concept of unconditionally closed and algebraic sets. Markov called a subset A of a group G unconditionally closed in G if it is closed in any Hausdorff group topology on G. Clearly, all solution sets of equations in G, as well as their finite unions and arbitrary intersections, are unconditionally closed. A subset A of a group G with identity element 1 is said to be elementary algebraic in G if there exists a word w =w(x) in the alphabet G U {x±1} (x is a variable) such that A = {x ∈G : w(x) = 1}. Finite unions of elementary algebraic sets are additively algebraic sets. An arbitrary intersection of additively algebraic sets is called algebraic. Thus, the algebraic sets in G are the solution sets of arbitrary conjunctions of finite disjunctions of equations. The concepts related to dimensions of metrizable groups, values of metrics, and free groups are also discussed in the chapter. Details of Maltsev spaces and retracts of groups are also presented in the chapter.