

September 3, 2014

O. Gutik


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The active participants of the seminar were discuss new results and posed some open problems.



September 10, 2014

O. Ravskyi


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The active participants of the seminar were discuss new results and posed some open problems.



September 17, 2014, October 1, 2014

O. Gutik


On closures in semitopological inverse semigroups with continuous inversion

We discuss on the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse
semigroups with continuous inversion. In particularly we show that a topological group G is Hclosed in the class
of semitopological inverse semigroups with continuous inversion if and only if G is compact, a Hausdorff linearly ordered
topological semilattice E is Hclosed in the class of semitopological semilattices if and only if E is Hclosed
in the class of topological semilattices, and a topological Brandt λ^{0}extension of S is
(absolutely) Hclosed in the class of semitopological inverse semigroups with continuous inversion if and only if so is S.
Also, we construct an example of an Hclosed semitopological semilattice in the class of semitopological semilattices which
is not absolutely Hclosed in the class of semitopological semilattices.



October 8, 2014, October 15, 2014

O. Ravskyi


On Hclosed quasitopological groups

We should prove that each Cauchy compete Hausdorff quasitopological group is Hclosed in the class of
Hausdorff quasitopological groups. In particular, a Hausdorff topological group G is Hclosed in the class of
Hausdorff quasitopological groups iff G is Raikov complete.



October 22, 2014, October 29, 2014

O. Ravskyi


Cowide topologies and their applications



November 12, 2014, November 19, 2014

O. Gutik, K. Mel'nyk


On the semigroup of cofinite momotone partial bijections of the real line

We discusss on the structure of the semigroup of cofinite momotone partial bijections of the real line.



November 26, 2014

O. Gutik


On Hclosed pospaces without infinite antichains

We describe the topology of an Hclosed pospace without infinite antichains and show that an Hclosed
pospace without infinite antichains is monotone normal.



December 3, 2014

O. Ravskyi


An introduction to Hclosed quasitopological groups

See the title. :) Also we show that a topological group G is Hclosed in the class of quasitopological
groups if and only if G is Raikov complete and present some examples.



December 29, 2014

S. Bardyla


On the kpolycyclic monoid

For every cardinal k we intriduce a notion of the kpolycyclic monoid
and study it's topological properties as a topological inverse semigroup.
For evere positive integer n we constract a minimal topology on the npolycyclic monoid and
prove that with this topology it is a Hcomplete topological inverse semigroup.



December 30, 2014

O. Chervak


On some asymptotic bounds for Ramsey numbers

We will discuss some new bounds for diagonal and offdiagonal Ramsey numbers.



January 13, 2015

T. Banakh


A natural uniformity on a paratopological group

We prove that each (Hausdorff) paratopological group G carries a natural
(separated) uniformity turning G into a uniform quasitopological
group.
This uniformity is generated by the base consisting of the sets {(x,y)∈
G×G: y∈ VxV^{1}∩V^{1}xV} where
V runs over neighborhoods of the unit in G. The presence of this
uniqformity guarantees that each Hausdorff paratopologial group G
condences onto a
Tychonoff quasitopological group and so G is functionally Hausdorff. If
G is countably Hausdorff, then G condences onto a metrizable
quasitopological group
and hence is submetrizable. This resolves two probelsm of Arkhangelski
and Tkachenko. More details can be found here.



January 16, 2015

S. Bardyla


On an embedding of the polycyclic monoid

We prove that for every non zero cardinal k the kPolycyclic monoid does not embed into any
countably compact topologycal semigroup.



February 11, 2015

O. Ravskyi


Divertisement

We started a new spring session of our seminar from the Divertisment. The active participants of the seminar
are discussed new results and they posed some open problems.



February 18, 2015

S. Bardyla


Some remarks on topologizations of the polycyclic monoid

We built locally compact non discrete topology Ƭ on the kpolycyclic monoid P such
that (P,Ƭ) is topologycal inverse semigroup. Also we proved, that Y\P is a ideal
for every semitopological semigroup Y such that cl(P)=Y, and construct a compact
topology Ʈ on P such that (P,Ʈ) is a semitopologycal semigroup.



February 25, 2015

I. Guran


On boundedness of topological rings

We consider various types of boundedness in topological rings: precompact, τnarrow, ets. Cardinal functions are investigated.



March 4, 2015

O. Gutik


On some problems on topological graph semigroups and infinite word semigroups

Some problems on topological graph semigroups and infinite word semigroups were posed.



March 11, 2015

I. Guran


On boundedness of topological rings

We consider various types of boundedness in topological rings: precompact, τnarrow, ets. Cardinal functions are investigated.



March 25, 2015

S. Bardyla


On a completion of quasitopological grups

We consider a completion G* of a quasitopologycal group G and prove that G* is an Hclosed
quasitopologycal group. Moreover G is a dense subgroup of G*.



April 1, 2015

O. Ravskyi


Characteristic subsets of locally compact abelian topological groups

A subset $A$ of a locally compact abelian topological group $G$ we shall call *characteristic*, if there is no sequence
$\{\phi_k\}$ of nontrivial characters of the group $G$ such that $\{\phi_k(a)\}$ converges to the unit for each element $x\in A$.
Investigating the problem which subsets $A$ are characteristic, we obtain some beginning results and pose some problems and directions of investigation.



April 7, 2015

S. Bardyla


On a completion of quasitopological grups

We consider a completion G* of a quasitopologycal group G and prove that G* is an Hclosed quasitopologycal
group. Moreover G is a dense subgroup of G*.



April 15, 2015

I. Guran


On boundedness of topological rings

We consider various types of boundedness in topological rings: precompact, τnarrow, ets. Cardinal functions are investigated.



April 22, 2015

S. Bardyla,
O. Ravskyi


Про наболіле та вимучене. Bardyla Theorem.

Епіграф: "Видел Ленин пашни, избы деревень,
видел он вчерашний и грядущий день!"
(Це  не реклама комунізму, це  ностальгія за недоведеною Серьоженькою теоремою!)
Inspired and filling the small gap by the first speaker, he and the second speaker proved that if (G,τ) is a Cauchy full
quasitopological group, (G,σ) is a quasitopological group, σͻτ, and the space (G,σ) has
a πbase, consisting of open in the space (G,τ) subets then (G,σ) is a Cauchy full quasitopological group too.



May 6, 2015, May 13, 2015

T. Banakh


kSpaces, their generalizations and applications

We consider some generalizations of the kspace property and discuss
their applications to free constructions of topological algebra.



June 3, 2015, June 10, 2015

L. Karchevska,
T. Banakh


On metrization of functors

We discuss the problem of metrization of functional functors and functors with finite supports.



June 17, 2015

T. Banakh


Cardinal invariants distinguishing permutation groups

We prove that for infinite cardinals κ<λ the alternating group Alt(λ) (of even permutations) of
λ is not embeddable into the symmetric group Sym(κ) (of all permutations) of κ. To prove this fact we introduce
and study several monotone cardinal group invariants which take value κ on groups Alt(κ) and Sym(κ).



June 18, 2015

S. Bardyla


On a completeness of factorizable topological inverse semigroups

We study a completeness of factorizable topological inverse semigroups and
introduce a method how to construct such semigroups.



June 30, 2015

А. Каток


Некоторые связи между топологией и динамикой

Мы рассмотрим некоторые связи между топологией дифференцируемого
многообразия и свойствами индивидуальных диффеоморфизмов и групп
диффеоморфизмов этого многообразия. Классические результаты такого
рода относятся к свойствам периодических орбит. Примеры: формула следа
Лефшеца, теория Нильсена. С развитием гиперболической динамики в
последние полвека появились результаты, которые гарантируют
существование сложных инвариантных множеств с весьма нетривиальной
динамикой. В последние 10 лет обнаружились весьма удивительные
свойства коммутативных групп диффеоморфизмов ранга больше единицы.
Необходимые понятия из теории динамических систем будут объяснены.



July 6, 2015

O. Gutik


On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero

We show that that if τ be a Hausdorff locally compact topology on the bicyclic semigroup with adjoined zero
C^{ 0} = C(p,q)∪{0} such that (C^{ 0},τ)
is a semitopological semigroup, then τ is either compact or discrete.



July 14, 2015

O. Равський


Нові пригоди коня та віслючка

Віслючок розкаже про свою важку долю без коня. Він надибав яблучко та
усе від нього відкушує, відкушує, але ніяк не проковтне. Віслючок тягнеться до
яблучка та підстрибує, але треба коня, бо у мене не кінські сили, а лише віслючі.



July 20, 2015

O. Gutik, K. Melnyk


On the structure of the semigroup of cofinite momotone partial homeomorphisms of the real line

We describe the structure of the semigroup of cofinite momotone partial homeomorphisms of the real line.



July 28, 2015

В. Любашенко


Kтеорія: Qконструкція Квилена

Переповідається Qконструкція Квилена для точних категорій A.
По такій будується категорія QA і її класифікуючий простір BQA.
Вищі Kгрупи визначаються як гомотопічні групи BQA.



July 30, 2015, July 31, 2015, August 3, 2015

O. Ravskyi


On extent of weakly submetaLindelöf spaces

We discuss on the topic of the title.

