
Date:

Speaker:

Title (click on the title to see the Abstract):



26.05.14

M.Pylypovych
M.Simkiv


Generalizing Kuratowski Theorem about 14 sets

Recently, Shalitt and Willard showed that Kuratowski 14set theorem does not have an immediate generalization for multitopological spaces.
Situation changes if the topologies are comparable. For a multitopological space with comparable topologies
we prove the finiteness of number of possible sets obtained by operations of closure and complement.
We also determine the maximal possible number of different sets obtained by these operations in a multitopological space
and prove that this number is indeed achieved on some subset of a suitable ntopological space.
The sequence 2,14,126,1394,... of these cardinalities is not included in Online Encyclopedia of Integer sequences.
This is a joint result with T.Banakh, O.Chervak, T.Martynyuk, and A.Ravsky







19.05.14

T.Radul


Convex hulls of functional functors

We define and study the convexification of functional functors and discuss some related open problems.







05.05.14
12.05.14

T.Banakh


On difference sets in partitions of Gspaces and groups

Using idempotent quasiinvariant measures on groups we shall prove that for each partition
G=A_{1}∪...∪A_{n} of a group G there is a finite subset
F of cardinality F≤n in G such that
G=FA_{i}A_{i}^{1}A_{i} for some cell of the partition.
This gives a partial answer to a problem of I.V.Protasov posed in 1995
in the Kourovka Problem Notebook.







7.04.14

M.Zarichnyi


On free Banach lattices

We consider some categorial properties of the construction of pairs of convex sets in Banach spaces
and prove that it leads to the construction of a free Banach lattice over a Banach space.







31.03.14

T.Banakh


On convergent sequences in scattered compact semilattices

We shall prove that for any point x of countable scattered height in a compact
topological semilattice S, any sequence (x_{n})_{n∈ω} that converges to x and any monotone sequences (x_{n,m})_{m∈ω} that converge to points x_{n}
there are a function f:ω→ω and an infinite subset A of ω such that
the diagonal subsequence (x_{n,f(n)})_{n∈A} converges to x.
Applying this theorem we give a simple proof of a difficult theorem of Banakh, Gutik and Rajagopalan who
proved in 2005 that each point of scattered height ≤ 2 in a compact topological semilattice
is the limit of a convergent sequence of isolated points of S.







24.03.14

T.Banakh


Canonical subsemilattices in scattered compact semilattices of low scattered height

We shall prove that each locally minimal point of scattered height 2
in a compact topological semilattice
S is contained in a compact countable subsemilattice T that contain dense set
of isolated points of S and is topologically isomorphic to one of 7 canonical semilattices.
This is a joint work with R.Bonnet and W.Kubis.







03.03.14 17.03.14

T.Banakh


Attractors of contractive function systems on multimetric spaces

We prove counterparts of Banach and Matkowski Fixed Point Theorems for contractive selfmaps
of multimetric spaces and apply the obtained theorems to studying attractors of contractive function systems
on topologicals spaces. In particular, we prove a metrization theorem for
topologically contracting function systems.







24.02.14

All


День прощання з Героями Майдану







17.02.14

T.Radul


Fixed Point and Nash equilibrium Theorems for binary convexities

We prove counterparts of Kakutani Fixed Point Theorem and
Nash Equilibrium Theorem for binary convexities and apply the
obtained results to binary monads. As a corollary we obtain the KozhanZarichnyi Equilibrium Theorem for capacities.







10.02.14

All


Divertissement

Active participants of the seminar will present and discuss interesting open problems
from various branches of topology and its applications.







3.02.14

I.Hetman


Topological structure of the hyperspaces of closed convex subsets in normed spaces

Some principal results the Ph.D. Thesis of Ivan Hetman is discussed.








* * *


* * * Winter Holidays * * *







23.12.13 30.12.13

I.Guran


Boundedness in topological rings

We shall discuss various notions of boundedness in topological rings.
In particular, we prove that each compact topological ring is totally disconnected.







16.12.13

A.Ravsky


On semiregular semitopological groups, II

We consider the problem of continuity of the multiplication on
groups, close to being compact.







9.12.13

T.Radul


Tensor products versus monads

It is wellknown that each monadic functor admits a tenor product.
We shall discuss the existence of tensor products for functors which cannot completed to a monad.







25.11.13

A.Ravsky


On semiregular semitopological groups

We consider the problem of continuity of the multiplication on
groups, close to being compact.







18.11.13

M.Zarichnyi


The asymptotic property C

We shall discuss the asymptotic property C in the coarge category and detect some groups with this property.







11.11.13

B.Bokalo


On some classes of discontinuous functions

We discuss the relation between some classes spaces (scattered, fragmentable) and functions (scatteredly continuous,
weakly discontinuous, feebly continuous, etc).
In particular, we prove that a fragmentable compact Hausdorff space is metrizable if and only if it is hereditarily Lindelof.







4.11.13

T.Banakh


Pytkeev ℵ_{0}spaces

We introduce a new class of generalized metric spaces consisting of Pytkeev ℵ_{0}space, i.e., a regular topological space
possessing a countable Pytkeev network. The class of Pytkeev ℵ_{0}spaces contains all metrizable separable spaces and
is (properly) contained in the class of ℵ_{0}spaces. This class is closed under many operations over
topological spaces: taking a subspace, a countable Tychonoff product, a countable small boxproduct, a direct limit topology,
taking the hyperspace and a function space endowed with the compactopen topology. More precisely, for every Hausdorff ℵ_{0}space X and
a Pytkeev ℵ_{0}space Y the function space C_{k}(X,Y) endowed with the compactopen topology is a Pytkeev
ℵ_{0}space. This enforces a classical result of E.Michael (1966) on function spaces between ℵ_{0}spaces.
Pytkeev ℵ_{0}spaces can be used
to get a new characterization of second countable spaces: a topological space is second countable if and only if it is a Pytkeev
ℵ_{0}space with countable fan tightness.







29.10.13

O.Chervak


Kuratowski 14set problem for multitopological spaces.

Recently, Shalitt and Willard showed that Kuratowski 14set theorem does not have an immediate generalization for multipological spaces.
Situation changes if topologies are comparable. For a multitopological space with comparable topologies we prove the finiteness of number of possible sets obtained by operations of closure and complement.
We also determine the maximal possible number of different sets obtained by these operation in a multitopological space.
Sequence 2,14,126,1394, of these cardinalities is not included in Online Encyclopedia of Integer sequences.
This is a joint result with T.Banakh, T.Martynyuk, M. Pylypovych, A.Ravsky







21.10.13

I.Pastukhova


Automatic continuity of homomorphisms between topological semigroups

In 1976 Yeager proved that a homomoprhism h:X→Y between compact topological Clifford semigroups is continuous if and only if
for any subsemilattice E and any subgroup H of X the restrictions hE and hH are continuous. In this talk we shall discuss
possible extensions of this Yeager's theorem beyond the class of compact topological Clifford semigroups.
This is a joint result with T.Banakh.







30.09.13

T.Banakh


Metrizability of Lawson compact preClifford semigroups

A semigroup S is called preClifford if it is a union of groups.
A topological preClifford semigroup S is called Lawson if for each idempotent e of S and its neighborhood O(e) in S there is an open
subsemigroup U of S which contains the maximal subgroup π^{1}(e) and is contained in the open set π^{1}(O(e)).
Here π:S→ E, π:x→xx^{1}=x^{1}x, is the projection of S onto the set E of idempotents of S.
It is proved that a Lawson preClifford compact topological semigroup S is metrizable if and only if the set E of idempotents of S is metrizable
and each maximal subgroup H_{e}=π^{1}(e), e in E, is metrizable.
This is a joint result with I.Pastukhova and O.Potyatynyk.







28.09.13

All


Топологічна прогулянка

В суботу 28 вересня у парку "Знесіння" (біля озера) відбулася традиційна топологічна прогулянка активних
учасників топологічного семінару (T.Банах, O.Равський, Т.Мартинюк) та гостей семінару
(A.Bartoszewicz, Sz.Glab, F.Strobin, M.Bienias) з Politechniki Lodzkiej.







23.09.13

O.Gutik


Primitive inverse pseudocompact (semi)topological semigroups

The speaker presented some results on structure of primitive inverse pseudocompact (semi)topological semigroups.







16.09.13

O.Ravsky


On 2pseudocompact paratopological groups which are topological groups

It is proved that a 2pseudocompact Hausdorff paratopological group is a topological group
if it contains a compact subset with countable pseudocharacter.







9.09.13

All


Divertissement

Active participants of the seminar will present and discuss interesting open problems
from various branches of topology and its applications.





