Igor Guran - Academia.edu (original) (raw)

Papers by Igor Guran

1. Lviv mathematical school and doctorants in mathematics in 1920–1939. Considering the history o... more 1. Lviv mathematical school and doctorants in mathematics in 1920–1939. Considering the history of research in mathematics at the Lviv University, one often pays attention for the period 1919–1939. Then a creative team of mathematicians was formed in Lviv. In the history of science, it is known as the Lviv mathematical school. Stefan Banach and Hugo Steinhaus were undisputed leaders of this team. The extent and level of research of the Lviv mathematical school in the 30-ies were comparable with those of the Gottingen school. Here is how it was written in a letter to professors of mathematics and natural sciences department to the Ministry of Education and religious confessions in 1933; the letter was signed by the Dean S. Banach ([1]): “One can safely say that no Polish university, and perhaps not many foreign universities organized teaching mathematics as thoroughly and efficiently as did the University of Lviv . . . ” “In recent years the number of announced by the Lviv school pro...

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2020

We discuss various modifications of separability, precompactness and narrowness in topological gr... more We discuss various modifications of separability, precompactness and narrowness in topological groups and test those modifications in the permutation groups S(X) and S<ω(X).

Open Problems in Topology II, 2007

Topology and its Applications, 2012

Applied General Topology, 2013

We prove that a Hausdorff paratopological group G is meager if and only if there are a nowhere de... more We prove that a Hausdorff paratopological group G is meager if and only if there are a nowhere dense subset A ⊂ G and a countable set C ⊂ G such that CA = G = AC.

We provide an outline of Stanisław Ulam’s results obtained in the framework of the widely underst... more We provide an outline of Stanisław Ulam’s results obtained in the framework of the widely understood Lvov school of mathematics.

We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative... more We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative binary operation is orientable. This implies that the M\"obius band admits no cancellative continuous binary operation. This answers a question posed by the second author in 2010.

Topology and its Applications

Visnyk Lvivskogo Universytetu. Seriya Mekhaniko-Matematychna

This is the list of open problems in topological algebra posed on the conference dedicated to the... more This is the list of open problems in topological algebra posed on the conference dedicated to the 20th anniversary of the Chair of Algebra and Topology of Lviv National University, that was held on 28 September 2001.

Topological Algebra and its Applications, 2013

In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in... more In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in each Hausdorff shift-invariant topology. The class of perfectly supportable semigroups includes each semigroup S such that FSym(X) ⊂ S ⊂ FRel(X) where FRel(X) is the semigroup of finitely supported relations on an infinite set X and FSym(X) is the group of finitely supported permutations of X. a∈S supt(a) ⊂ X is called the support of S. A typical example of a supt-semigroup is the group Sym(X) of all bijections f : X → X of a set X, endowed with the support map supt : f → {x ∈ X : f (x) = x}. In Section 4 we shall describe another supt-semigroup Rel(X), which contains Sym(X) (and many other semigroups) as a supt-subsemigroup. Definition 2.1 implies the following proposition-definition.

Topology and its Applications, 2003

We construct a separable metrizable countable-dimensional (respectively strongly countable-dimens... more We construct a separable metrizable countable-dimensional (respectively strongly countable-dimensional) abelian topological group that contains all countable-dimensional separable metrizable abelian topological absolute Gδ-groups (respectively all finite-dimensional separable metrizable abelian topological absolute Gδσ-groups). A similar result is proved for separable metrizable abelian topological group that are countable-dimensional with respect to integral cohomological dimension.

Topological Algebra and its Applications

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch ... more A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.