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Introduction
I am from Lviv topological school (http://www.franko.lviv.ua/faculty/mechmat/Departments/Topology/trees.html). Nevertheless, I have publications in theory of (para)topological groups, general topology, combinatorial geometry, combinatorics, graph theory, algebra, and functional analysis. All of them are free, most of them are available online. I do not upload or update my papers at Research Gate. Almost all my recent papers are in arXiv: http://arxiv.org/find/math/1/au%3a+Ravsky/0/1/0/all/0/1#!
Additional affiliations
November 2004 - present
Education
November 1998 - May 2001
September 1993 - June 1998
Publications
Publications (88)
UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn...
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the smallest number of lines in $\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, th...
We study the following combinatorial problem. Given a set of n y-monotone curves, which we call , a determines the order of the wires on a number of horizontal such that any two consecutive layers differ only in swaps of neighboring wires. Given a multiset L of (that is, unordered pairs of wires) and an initial order of the wires, a tangle L if eac...
Let $G$ be a paratopological group. Following F. Lin and S. Lin, we say that the group $G$ is pseudobounded, if for any neighborhood $U$ of the identity of $G$, there exists a natural number $n$ such that $U^n=G$. The group $G$ is $\omega$-pseudobounded, if for any neighborhood $U$ of the identity of $G$, the group $G$ is a union of sets $U^n$, whe...
A subset S of a paratopological group G is a suitable set for G, if S is a discrete subspace of G, \(S\cup \{e\}\) is closed, and the subgroup \(\langle S\rangle \) of G generated by S is dense in G. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general investigation of...
Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a $k$-space $X$ its free semitopological $R$-module is a topological $R$-module. Also we construct a Tychonoff...
We discuss various modifications of separability, precompactness and narrowness in topological groups and test those modifications in the permutation groups S(X) and \(S_{<\omega }(X)\).
A topological group X is called duoseparable if there exists a countable set S⊆X such that SUS=X for any neighborhood U⊆X of the identity. We construct a functor F assigning to each (abelian) topological group X a duoseparable (abelian-by-cyclic) topological group FX, containing an isomorphic copy of X. In fact, the functor F is defined on the cate...
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X\,=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.
p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of
countably pracompact topological spaces. We construct a pseudocompact topologi...
We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each 2-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-compact feebly compact paratopological...
Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space fro...
An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given...
A paratopological group $G$ has a {\it suitable set} $S$. The latter means that $S$ is a discrete subspace of $G$, $S\cup \{e\}$ is closed, and the subgroup $\langle S\rangle$ of $G$ generated by $S$ is dense in $G$. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general...
Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
Let κ be an infinite cardinal. A topological space X is κ-bounded if the closure of any subset of cardinality ≤κ in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) κ-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a s...
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 c...
We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergen...
Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from...
Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
We study the following combinatorial problem. Given a set of $n$ y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of wir...
We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<\omega}(X)$.
A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the...
A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, bu...
An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given...
The segment number of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for planar g...
We study the following combinatorial problem. Given a set of n y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset L of swaps (that is, unordered pairs of numbers between 1 and n) and...
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.
A topologized semilattice X is complete if each non-empty chain C⊂X has infC∈C¯ and supC∈C¯. It is proved that for any complete subsemilattice X of a functionally Hausdorff semitopological semilattice Y the partial order ≤X={(x,y)∈X×X:xy=x} of X is closed in Y×Y and hence X is closed in Y. This implies that for any continuous homomorphism h:X→Y f...
The \emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for p...
We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of $\omega{\times\omega}$-matrix units cannot be embedded into a topological semigroup which is an...
The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed...
In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact t...
We discuss the problem of embeddibility of a topological space into a Hausdorff $\omega$-bounded space, and present two canonical constructions of such an embedding.
A subset $S$ of an Abelian group is $decomposable$ if $ \emptyset\ne S\subset S+S$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $S$ of an Abelian group contains a non-empty subset $T\subset S$ with $\sum T=0$. For every $n\in\mathbb N$ we present a decomposable subset $S$ of cardinality $|S...
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 c...
We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of...
We prove that any topological group $G$ containing a subspace $X$ of the Sorgenfrey line has spread $s(G)\ge s(X\times X)$. Under OCA, each topological group containing an uncountable subspace of the Sorgenfrey line has uncountable spread. This implies that under OCA a cometrizable topological group $G$ is cosmic if and only if it has countable spr...
A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular) banalytic space has countable spread (and under PFA is hereditarily Lindel\"of). Applying banalytic spaces to top...
We study the following combinatorial problem. Given a set of $n$ y-monotone wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of numbers between 1 and $n$...
An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges into \emph{bundles} when they travel in parallel. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a \emph{bundled crossing}. We minimize the number of bundled crossings. We conside...
A topologized semilattice $X$ is complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. It is proved that for any complete subsemilattice $X$ of a functionally Hausdorff semitopological semilattice $Y$ the partial order $P=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$...
We introduce three new classes of pracompact spaces, consider their basic properties and relations with other compact-like spaces.
We introduce three new classes of pracompact spaces, consider their basic properties and relations with other compact-like spaces.
Generalizing the famous 14-set closure-complement Theorem of Kuratowski from
1922, we prove that for a set $X$ endowed with $n$ pairwise comparable
topologies $\tau_1\subset\dots\subset\tau_n$, by repeated application of the
operations of complement and closure in the topologies $\tau_1,\dots,\tau_n$ to
a subset $A\subset X$ we can obtains at most...
An $H$-closed quasitopological group is a Hausdorff quasitopological group
which is contained in each Hausdorff quasitopological group as a closed
subspace. We obtained a sufficient condition for a quasitopological group to be
$H$-closed, which allowed us to solve a problem by Arhangel'skii and Choban and
to show that a topological group $G$ is $H$...
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right visualizing progression of time. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual...
Given a drawing of a graph, its visual complexity is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. introduced a different measure for the visual complexity, the affine cover number, which...
Given a drawing of a graph, its \emph{visual complexity} is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. [GD 2016] introduced a different measure for the visual complexity, the \emph{affi...
It is well known that any graph admits a crossing-free straight-line drawing in \(\mathbb {R} ^3\) and that any planar graph admits the same even in \(\mathbb {R} ^2\). For a graph G and \(d \in \{2,3\}\), let \(\rho ^1_d(G)\) denote the minimum number of lines in \(\mathbb {R} ^d\) that together can cover all edges of a drawing of G. For \(d=2\),...
In the paper we study the preservation of pseudocompactness (resp., countable
compactness, sequential compactness, $\omega$-boundedness, totally countable
compactness, countable pracompactness, sequential pseudocompactness) by
Tychonoff products of pseudocompact (and countably compact) to\-pological
Brandt $\lambda_i^0$-extensions of semitopologica...
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just...
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just...
In their seminal work, Mustafa and Ray (2009) showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search -- this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via...
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can accommodate all edges of a drawing of $G$, where $\rho^1_2(G)$ is defined for p...
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawi...
Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free
topological group and the free Abelian topological group over $X$ in the sense
of Markov. In this paper, we discuss two topological properties in $F(X)$ or
$A(X)$, namely the countable tightness and $\mathfrak G$-base. We provide some
characterizations of the countable tigh...
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.
We prove that a regular topological space $X$ is Tychonoff if and only if its
topology is generated by a quasi-uniformity $\mathcal U$ such that for every
$U\in\mathcal U$ and $A\subset X$ there is $V\in\mathcal U$ such that $B(\bar
A,V)\subset \bar{B(A,U)}$. This characterization implies that each regular
paratopological group is Tychonoff and eac...
For any topological space $X$ we study the relation between the universal
uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and
the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity
$\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$
we define the verbal power $\mathcal U^v$ of $...
We derive many upper bounds on the submetrizability number and $i$-weight of
paratopological groups and topological monoids with open shifts. In particular,
we prove that each first countable Hausdorff paratopological group is
submetrizable thus answering a problem of Arhangelskii posed in 2002. Also we
construct an example of a zero-dimensional (a...
We define a notion of a rotund quasi-uniform space and describe a new direct
construction of a (right-continuous) quasi-pseudometric on a (rotund)
quasi-uniform space. This new construction allows to give alternative proofs of
several classical metrizability theorems for (quasi-)uniform spaces and also
obtain some new metrizability results. Applyin...
We introduce so-called cone topologies of paratopological groups, which are a
wide way to construct counterexamples, especially of examples of compact-like
paratopological groups with discontinuous inversion. We found a simple
interplay between the algebraic properties of a basic cone subsemigroup S of a
group G and compact-like properties of two b...
We obtain necessary and sufficient conditions when a pseudocompact
paratopological group is topological. (2-)pseudocompact and countably compact
paratopological groups that are not topological are constructed. It is proved
that each 2-pseudocompact paratopological group is pseudocompact and that each
Hausdorff \sigma-compact pseudocompact paratopol...
We study structure of inverse primitive pseudocompact semitopological and topological semigroups. We find the conditions when the maximal subgroup of an inverse primitive pseudocompact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Also an analogue of Co...
Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of
$X$, we prove that for every partition $X=A_1\cup\dots\cup A_n$ of $X$ into
$n\ge 2$ pieces there is a piece $A_i$ of the partition and a finite set
$F\subset G$ of cardinality $|F|\le
\phi(n+1):=\max_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$ such that $G=F\cdot
\Delta(A_i)$ wher...
We prove that a topological Clifford semigroup $S$ is metrizable if and only
if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is
a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also
for any countably compact Clifford topological semigroup $S$.
If K' and K are convex bodies of the plane such that K' is a subset of K then
the perimeter of K' is not greater than the perimeter of K. We obtain the
following generalization of this fact. Let K be a convex compact body of the
plane with the perimeter p and the diameter d and r>1 be an integer. Let s be
the smallest number such that for any curve...
We found a solution of the star puzzle (a path on a chessboard from c5 to d4
in 14 straight strokes) in 14 queen moves, which has been claimed by the author
as impossible.
A convex subset X of a linear topological space is called compactly convex if
there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that
$[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex
subset of the plane is compactly convex. On the other hand, the space $R^3$
contains a convex set that is not compactly co...
Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the vertex set of G into the plane. Let be the maximum integer k such that there exists a crossing-free redrawing π′ of G which keeps k vertices f...
We consider approximations of a continuous function on a countable normed
Fr\'{e}chet space by analytic and $*$-analytic. Also we found a criterium of
the existence of an extension of a continuous function from a dense subspace of
a topological space onto the space.
We prove a precise formula for the minimal number K(n) such that every binary
word of length $n$ can be divided into K(n) palindromes. Also we estimate the
average number $\ol K(n)$ of palindromes composing a random binary word of the
length n.
A binary word is symmetric if it is a palindrome or an antipalindrome. We define a new measure of asymmetry of a binary word equal to the minimal number of letters of the word whose deleting from the word yields a symmetric word and obtain upper and lower estimations of this measure. Comment: 4 pages
We consider the Steinhaus geometrical game on cake dividing. Hugo Steinhaus in his popular book [One hundred problems in elementary mathematics (Pergamon Press, Oxford) (1963; Zbl 0116.24102)] (see also B. Grünbaum [Studies in combinatorial geometry and the theory of convex bodies (Nauka Moskau) (1971; Zbl 0229.52001)]) considered the following gam...
A paratopological group $G$ is saturated if the inverse $U^{-1}$ of each non-empty set $U\subset G$ has non-empty interior. It is shown that a [first-countable] paratopological group $H$ is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if $H$ admits a continuous bijective homomorphism onto a (totally...
A Hausdorff paratopological group G is H-closed if G is closed in each Hausdorff paratopological group containing G. We obtain criteria of H-closedness for some classes of abelian paratopological groups. In particular, for topological groups. Comment: 7 pages
Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the quotient $G/H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^\flat$. On the other hand, we...
We prove that a Hausdorff paratopological group G is meager if and only if
there are a nowhere dense subset A of G and a countable subset C in G such that
CA=G=AC.
We introduce and study oscillator topologies on paratopological groups and
define certain related number invariants. As an application we prove that a
Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology
provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating
(resp. 2-oscillating) provided for any neighbo...
We consider straight-line drawings of a planar graph G with possible edge crossings. The untangling problem is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let fix
G denote the maximum number of vertices that can be left fixed in the worst case among all drawings of G. In the allocation problem, we are giv...
Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the vertex set of G into the plane. Let fix(G, π) be the maximum integer k such that there exists a crossing-free redrawing π ′ of G which keeps k...
A binary word is symmetric if it is a palindrome or an antipalindrome. We define a new measure of asymmetry of a binary word equal to the minimal number of letters of the word whose deleting from the word yields a symmetric word and obtain upper and lower estimations of this measure.
We prove a precise formula for the minimal number K(n) such that every binary word of length n can be divided into K(n) palindromes. Also we estimate the average number K(n) of palindromes composing a random binary word of the length n.