Unconditionally τ-closed and τ-algebraic sets in groups

Families of unconditionally tau-closed and tau-algebraic sets in a group are defined, which are natural generalizations of unconditionally closed and algebraic sets defined by Markov. A sufficient condition for the coincidence of these families is found. In particular, it is proved that these families coincide in any group of cardinality at most tau. This result generalizes both Markov's theorem on the coincidence of unconditionally closed and algebraic sets in a countable group (as is known, they may be different in an uncountable group) and Podewski's theorem on the topologizability of any ungebunden group. (c) 2007 Elsevier B.V. All rights reserved.