Unconditionally closed and algebraic sets in groups
by
Olga Sipacheva
Department of General Topology and Geometry, Mechanics and Mathematics Faculty, Moscow State University

A subset A of a group G is said to be unconditionally closed in G if it is closed in any Hausdorff group topology on G (A. A. Markov, 1945). Note that a group G is nontopologizable (i.e., the only Hausdorff group tology which G admits is descrete) if and only if the complement of the identity element is unconditionally closed in G.

Clearly, all solution sets of equations in G, as well as their finite unions and arbitrary intersections, are unconditionally closed. Such sets are called algebraic. Markov's precise definition is as follows.

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∪{x^±1} (x is a variable) such that

A = {x ∈ G: w(x) = 1}.

Finite unions of elementary algebraic sets are called additively algebraic sets. An arbitrary intersection of additively algebraic sets is said to be algebraic. Thus, the algebraic sets in G are the solution sets of arbitrary conjunctions of finite disjunctions of equations.

In 1945, Markov showed that any algebraic set is unconditionally closed and posed the problem of whether the converse is true. Moreover, he proved that any unconditionally closed set in a countable group is algebraic. In 1979, Hesse constructed an example in his PhD dissertation showing that the answer to the general question is no. We show that Shelah's CH example of a nontopologizable group, which has amazing additional properties, also provides an example.

An obvious necessary condition for the topologizability of a group G is that no finite system of inequations in G can have a unique solution (i.e., the complement to an additively algebraic set cannot be one-point). In 1977, Podewski suggested a natural sufficient condition that no system of fewer than |G| inequations has a unique solution; he called groups with this property ungebunden.

We discuss sufficient conditions for the coincidence of unconditionally closed and algebraic sets. We also introduce families of unconditionally t-closed and t-algebraic sets in a group, which are natural generalizations of unconditionally closed and algebraic sets defined by Markov, and find a sufficient condition for the coincidence of these families. In particular, we show that these families coincide in any group of cardinality at most t. This result generalizes both Markov's theorem on the coincidence of unconditionally closed and algebraic sets in a countable group and Podewski's theorem on the topologizablity of any ungebunden group.

We also touch upon the retationship between the notion of hereditary topologizability of groups (introduced recently by Gabor Lukacs) and that of categorical compactness, or c-compactness of groups (indroduced by Dikranjan and Uspenskij), which was first noticed by Lukacs.

Date received: May 1, 2008