Unconditionally tau-Closed and tau-Algebraic Sets in Groups

Details Unconditionally tau-Closed and tau-Algebraic Sets in Groups Unconditionally tau-Closed and tau-Algebraic Sets in Groups

2007-03-13

arxiv.org/abs/math/0703397v3

Mathematics

Group Theory

Version 2: A mistake is corrected. The main result is changed accordingly Version 3: Minor changes are made

Scientific paper

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 topologizablity of any ungebunden group.

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