Unconditionally tau-Closed and tau-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 Markovs 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 Podewskis theorem on the topologizablity of any ungebunden group.