# Protasov I. V.

### Descriptive complexity of the sizes of subsets of groups

Banakh T. O., Protasov I. V., Protasova K. D.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1280-1283

We study the Borel complexity of some basic families of subsets of a countable group (large, small, thin, rarefied, etc.) determined by the sizes of their elements. The obtained results are applied to the Czech – Stone compactification $\beta G$ of the group $G$. In particular, it is shown that the closure of the minimal ideal $\beta G$ has the $F_{\sigma \delta}$ type.

### Ultrafilters on balleans

Protasov I. V., Slobodianiuk S. V.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1698-1706

A ballean (equivalently, a coarse structure) is an asymptotic counterpart of a uniform space. We introduce three ultrafilter satellites of a ballean (namely, corona, companion, and corona companion), evaluate the basic cardinal invariants of the corona and characterize the subsets of balleans in terms of companions.

### Scattered Subsets of Groups

Banakh T. O., Protasov I. V., Slobodianiuk S. V.

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 304-312

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 ℵ_{0} scattered subsets.

### Thin Subsets of Groups

Protasov I. V., Slobodianiuk S. V.

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1245–1253

For a group *G* and a natural number m; a subset *A* of *G* is called m-thin if, for each finite subset *F* of *G*; there exists a finite subset *K* of *G* such that |*F* _{ g } ∩ *A*| ≤ *m* for all *g* ∈ *G* \ *K*: We show that each *m*-thin subset of an Abelian group *G* of cardinality ℵ_{ n }; *n* = 0, 1,… can be split into ≤ *m* ^{ n+1} 1-thin subsets. On the other hand, we construct a group G of cardinality ℵ_{ ω } and select a 2-thin subset of *G* which cannot be split into finitely many 1-thin subsets.

### Balleans and *G* -spaces

Petrenko O. V., Protasov I. V.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 344-350

We show that every ballean (equivalently, coarse structure) on a set $X$ can be determined by some group $G$ of permutations of $X$ and some group ideal $\mathcal{I}$ on $G$. We refine this characterization for some basic classes of balleans: metrizable, cellular, graph, locally finite, and uniformly locally finite. Then we show that a free ultrafilter $\mathcal{U}$ on $\omega$ is a $T$-point with respect to the class of all metrizable locally finite balleans on $\omega$ if and only if $\mathcal{U}$ is a $Q$-point. The paper is concluded with а list of open questions.

### Relatively thin and sparse subsets of groups

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 216-225

Let $G$ be a group with identity $e$ and let $\mathcal{I}$ be a left-invariant ideal in the Boolean algebra $\mathcal{P}_G$ of all subsets of $G$. A subset $A$ of $G$ is called $\mathcal{I}$-thin if $gA \bigcap A \in \mathcal{I}$ for every $g \in G \ \{e\}$. A subset $A$ of $G$ is called $\mathcal{I}$-sparse if, for every infinite subset $S$ of $G$, there exists a finite subset $F \subset S$ such that $\bigcap_{g \in F}gA \in F$. An ideal $\mathcal{I}$ is said to be thin-complete (sparse-complete) if every $\mathcal{I}$-thin ($\mathcal{I}$-sparse) subset of $G$ belongs to $\mathcal{I}$. We define and describe the thin-completion and the sparse-completion of an ideal in $\mathcal{P}_G$.

### On the Suslin Number of Totally-Bounded Left-Topological Groups

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1581-1573

For every infinite cardinal α, we construct a zero-dimensional totally-bounded left-topological group with Suslin number α.

### Morphisms of Ball's Structures of Groups and Graphs

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 847-855

We introduce and study two kinds of morphisms between ball's structures related to groups and graphs.

### Ultrafilters and Decompositions of Abelian Groups

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 85-93

We prove that every *PS*-ultrafilter on a group without second-order elements is a Ramsey ultrafilter. For an arbitrary *PS*-ultrafilter ϕ on a countable group *G*, we construct a mapping *f*: *G* → ω such that *f*(ϕ) is a *P*-point in the space ω*. We determine a new class of subselective ultrafilters, which is considerably wider than the class of *PS*-ultrafilters.

### Irresolvable left topological groups

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 758–765

We prove that an irresolvable left topological group is of the first category. The pseudocharacter of an irresolvable left topological group*G* is countable, provided that*G* is Abelian or its cardinality is nonmeasurable. Some other cardinal invariants of an irresolvable left topological group are also determined.

### Irresolvable topologies on groups

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1646–1655

We prove that there exist *ZFC* models in which every nondiscrete topological Abelian group can be decomposed into countably many dense subsets. This statement is an answer to the question raised by Comfort and van Mill. We also prove that every submaximal left-topological Abelian group is σ-discrete.

### Decompositions of direct products of groups

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1385–1395

We propose a new method for the decomposition of direct products of groups into U subsets. By using this method, we prove the following generalization of the Comfort-van Mill theorem: An arbitrary nondiscrete topological Abelian group with a finite number of second-order elements can be decomposed into a countable number of dense subsets.

### Ideals and free Pairs in the semigroup β ℤ

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 573–580

We prove that the equations ξ+*x*=*m*ξ+*y*, *x*+ξ=*y*+*m*ξ have no solutions in the semigroup β ℤ for every free ultrafilter ξ and every integer *m*∈0, 1. We study semigroups generated by the ultrafilters ξ, *m*ξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ℤ.

### Bourbaki spaces of topological groups

Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 542–549

### Ochan topologies on the space of closed subgroups

Protasov I. V., Stukotilov V. S.

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1337–1342

### Souslin number of the space of subgroups of a locally compact group

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 654-658

### Compact subspaces in the space of subgroups of a topological group

Ukr. Mat. Zh. - 1986. - 38, № 5. - pp. 600–605

### Topological groups with a σ-compact space of subgroups

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 93 – 98

### The Chabauty topology on the lattice of closed subgroups

Protasov I. V., Tsybenko Yu. V.

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 207 - 213

### Connectedness of the space of subgroups

Protasov I. V., Tsybenko Yu. V.

Ukr. Mat. Zh. - 1983. - 35, № 3. - pp. 382 — 385

### Topological properties of the lattice of subgroups

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 355 - 360

### Dualisms of topological Abelian groups

Ukr. Mat. Zh. - 1979. - 31, № 2. - pp. 207–211

### Projections of abelian topological groups

Ukr. Mat. Zh. - 1978. - 30, № 4. - pp. 551–556

### Topological dualities of locally compact Abelian groups

Ukr. Mat. Zh. - 1977. - 29, № 5. - pp. 625–631

### Groups with uniquely generated invariant subgroups

Ukr. Mat. Zh. - 1977. - 29, № 2. - pp. 275–280