Igor Protasov

Maximal resolvability of bounded groups

A new original method for proving resolvability of topological groups is decribed. With the aid of this method, maximal resolvability is proved for some classes of topological groups, in particular, for the class of totally bounded groups.

Scattered Subsets of Groups

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets.

Algebraically determined topologies on permutation groups

Abstract In this paper we answer several questions of Dikran Dikranjan about algebraically determined topologies on the groups S ( X ) (and S ω ( X ) ) of (finitely supported) bijections of a set X . In particular, confirming conjecture of Dikranjan, we prove that the topology T p of pointwise convergence on each subgroup G ⊃ S ω ( X ) of S ( X ) is the coarsest Hausdorff group topology on G (more generally, the coarsest T 1 -topology which turns G into a [semi]-topological group), and T p coincides with the Zariski and Markov topologies Z G and M G on G . Answering another question of Dikranjan, we prove that the centralizer topology T G on the symmetric group G = S ( X ) is discrete if and only if | X | ⩽ c . On the other hand, we prove that for a subgroup G ⊃ S ω ( X ) of S ( X ) the centralizer topology T G coincides with the topologies T p = M G = Z G if and only if G = S ω ( X ) . We also prove that the group S ω ( X ) is σ -discrete in each Hausdorff shift-invariant topology.

Topologizations of a set endowed with an action of a monoid

Abstract Given a set X and a family G of self-maps of X, we study the problem of the existence of a non-discrete Hausdorff topology on X with respect to which all functions f ∈ G are continuous. A topology on X with this property is called a G-topology. The answer is given in terms of the Zariski G-topology ζ G on X, that is, the topology generated by the subbase consisting of the sets { x ∈ X : f ( x ) ≠ g ( x ) } and { x ∈ X : f ( x ) ≠ c } , where f , g ∈ G and c ∈ X . We prove that, for a countable monoid G ⊂ X X , X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζ G is non-discrete; moreover, in this case, X admits 2 c hereditarily normal G-topologies.

CLASSIFYING HOMOGENEOUS CELLULAR ORDINAL BALLEANS UP TO COARSE EQUIVALENCE

For every ballean

SYNDETIC SUBMEASURES AND PARTITIONS OF G-SPACES AND GROUPS

X

Selective and Ramsey Ultrafilters on

we introduce two cardinal characteristics

G

cov^\flat(X)

-spaces

and

On asymorphisms of groups

cov^\sharp(X)