Two theorems on imbeddings of 0-dimensional groups

In the class of 0-dimensional groups with infinite weight, the universal group is constructed. We prove that a 0-dimensional group can be imbedded into a multiplicative subgroup of a topological ring.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1594–1596, November, 1994.     

Decompositions of direct products of groups

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. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1385–1395, October, 1997.     

Functionally balanced groups

Translated from Matematicheskie Zametki, Vol. 49, No. 6, 87–91, June, 1991.     

On linear selections of convex set-valued maps

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x ₁ + x ₂) ⊂ φ(x ₁) + φ(x ₂). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.     

Topologizations of G-spaces

For a G-space X, we put and explore a question whether X admits a non-discrete Hausdorff G-invariant topology.