Searcharxiv⌕ Search

arXiv subjects

Alex Ravsky

Publications and source records attributed to Alex Ravsky.

48 records · Page 3Linked to original sources
arXiv2015

Each regular paratopological group is completely regular

We prove that a semiregular topological space $X$ is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

math.GN↗
arXiv2015

On the submetrizability number and $i$-weight of quasi-uniform spaces and paratopological groups

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 (and hence regular) Hausdorff paratopological abelian group $G$ with countable pseudocharacter which is not submetrizable. In fact, all results on the $i$-weight and submetrizability are derived from more general results concerning normally quasi-uniformizable and bi-quasi-uniformizable spaces.

math.GN↗
arXiv2014

Quasi-pseudometrics on quasi-uniform spaces and quasi-metrization of topological monoids

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. Applying this construction to topological monoids with open shifts, we prove that the topology of any (semiregular) topological monoid with open shifts is generated by a family of (right-continuous) left-subinvariant quasi-pseudometrics, which resolves an open problem posed by Ravsky in 2001. This implies that a topological monoid with open shifts is completely regular if and only if it is semiregular. Since each paratopological group is a topological monoid with open shifts these results apply also to paratopological groups.

math.GN↗
arXiv2014

The regularity of quotient paratopological groups

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 construct an example of a regular abelian paratopological group $G$ containing a closed discrete subgroup $H$ such that the quotient $G/H$ is Hausdorff but not regular.

math.GR↗
arXiv2012

On curves contained in convex subsets of the plane

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 of length greater than s contained in K there is a straight line intersecting the curve at least in r+1 different points. Then s=rp/2 if r is even and s=(r-1)p/2+d if r is odd.

math.MG↗
arXiv2011

Metrizability of Clifford topological semigroups

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 $G_δ$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.

math.GN↗
arXiv2010

On the palindromic decomposition of binary words

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.

math.CO↗
arXiv2010

A new measure of asymmetry of binary words

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.

math.CO↗
arXiv2010

On subgroups of saturated or totally bounded paratopological groups

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 bounded) [abelian] topological group $G$ [such that for each neighborhood $U\subset H$ of the unit $e$ there is a closed subset $F\subset G$ with $e\in h^{-1}(F)\subset U$]. As an application we construct a paratopological group whose character exceeds its $π$-weight as well as the character of its group reflexion. Also we present several examples of (para)topological groups which are subgroups of totally bounded paratopological groups but fail to be subgroups of regular totally bounded paratopological groups.

math.GR↗
arXiv2010

On H-closed paratopological groups

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.

math.GR↗