

14.05.07.

T.Banakh


Symmetric subsets and partitions of the Lobachevsky plane.

We prove that for any partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset. This distinguishes the Lobachevsky plane from the Euclidean plane which can be divided into 3 Borel pieces containing no unbounded centrally symmetric subset.



07.05.07

T.Radul


Hyperspace as intersection of idempotent measures and inclusion hyperspaces.



23.04.07.

M.Zarichnyi


Milytin maps for probability and idempotent measures.



16.04.07.

T.Banakh


Compactly convex sets.

A convex subset C of a linear topological
space is called
compactly convex if there is a continuous
compactvalued map
F assigning to each point x of C a compact subset F(x)
of C so that [x,y]
lies in the union of F(x) and F(y). We prove that each
convex subset of the plane is
compactly convex. On the other hand, the 3dimensional
space
R^{3} contains a convex subset which is
not compactly convex.
This is a joint work with M.Mitrofanov and O.Ravsky.



02.04.07.

T.Banakh


Continuously homogeneous spaces.

A topological space X is called continuously homogeneous if there is a homeomorphism H of X^{3} such that H(x,y,z)=(x,y,h_{x,y}(z)) and H(x,y,x)=(x,y,y) for all points x,y,z in X.
It is clear that each topological group is continuously homogeneous with the homeomorphism H(x,y,z)=(x,y,yx^{1}z). An example of a continuously homogeneous space homeomorphic to no topological group is the 7dimensional sphere S^{7}. On the other hand, the Hilbert cube fails to be continuously homogeneous.
This is joint work with Z.Kosztolowicz (Kilece).



26.03.07.

T.Banakh


Groupsequential topological groups.

A topological group G is called groupsequential if it carries the strongest group topology inducing the original topology on each convergent sequence. We give examples of such topological groups, characterize groupsequential free topological groups and pose some open problems.



12.03.07.

O.Gutik


Hclosed topological semigroups and semilattices.

In the report we shall discuss about some properties of Hclosed topological semigroups and semilattices.



05.03.07.

K.Koporkh


On spaces of quotient maps.



26.02.07.

M.Zarichnyi


Spaces of uppersemicontinuous capacities.

The notion of capacity was introduced into mathematics by G.Choquet. The family of all uppersemicontinuous capacities is endowd with the weak* topology. The obtained construction is functorial in the category of compact Hausdorff spaces and the talk is devoted to properties of this functor. In particular, the open mapping theorem will be proved.



25.12.06.

T.Radul


Openess of HartmanMycielski functor.

We investigate topological properties of HartmanMycielski functor on non metrizable compacta.



18.12.06.

O.Shukel'


Functors of finite degree and asymptotic dimension zero.

For any finitary normal functor F in the category of compact Hausdorff spaces one can define its counterpart on the category of proper metric spaces and coarse maps. The aim of this talk is to show that the obtained functor preserves the class of proper metric spaces of asymptotic dimension zero in the sense of Gromov. The obtained result is a counterpart of the corresponding result of Basmanov in the category of compact Hausdorff spaces.



11.12.06.

T.Banakh


Tranfinite separation dimension.



04.12.06.

M.Zarichnyi


Idempotent probability measures.

The set of all idempotent probability measures (Maslov measures) on
a compact Hausdor. space endowed with the weak* topology determines is func
torial on the category Comp of compact Hausdor. spaces. We prove that the
obtained functor is normal in the sense of E. Shchepin. Also, this functor is the
functorial part of a monad on Comp. We prove that the idempotent probability
measure monad contains the hyperspace monad as its submonad. A counterpart
of the notion of Milyutin map is de.ned for the idempotent probability measures.
Using the fact of existence of Milyutin maps we prove that the functor of idem
potent probability measures preserves the class of open surjective maps. Unlikely
to the case of probability measures, the correspondence assigning to every pair of
idempotent probability measures on the factors the set of measures on the product
with these marginals, is not open.



20.11.06.

Т.Banakh


On scatteredly continuous maps between topological spaces.

A bijective map $h:X\to Y$ between topological
spaces is a {\em scattered homeomorphism} if both $h$ and $h^{1}$
are scatteredly continuous. A map $f:X\to Y$ between topological
spaces is defined to be {\em scatteredly continuous} if for each
subspace $A\subset X$ the restriction $fA$ has a point of
continuity. We characterize scatteredly continuous maps in various terms and also study properties of topological spaces preserved by
scatteredly continuous maps and scattered homeomorphisms.



15.11.06.

Szymon Głąb


On Komjath property of sigmaideals.



15.11.06.

Alexander Balinsky


Zero modes of Pauli operator



13.11.06.

P. BorodulinNadzieja


Minimally generated Boolean algebra.



06.11.06.

Т.Banakh


On scattered compactifications of scattered metrizable spaces.

We prove that each metrizable scattered space has a hereditarily paracompact scattered compactification. Also we show that the class of hereditarily paracompact scattered compact spaces coincides with the smallest class K of compacta closed with respect to taking one point compactification of discrete topological sums of compacta from K.
Consequently each compact of the class K is uniform Eberlein.



30.10.06.

E.Tymchatyn


Convex metrics in the noncompact setting.

Bing (1949) showed that every Peano continuum X admits an
equivalent convex metric. To do this he showed
first that every Peano continuum can be partitioned into finitely
many small Peano continuous each pair of which meet only in their
common boundary.
We prove that if an arcwise connected and connected metric space
X admits a vanishing sequence of partitions U_{i} with U_{i+1} refining
U_{i}, then X admits an
equivalent convex metric. E.g., Plane with all points (x,y) where
x and y are both irrational removed.



25.10.06.

E.Tymchatyn


Milutin maps and topological uniqueness of fibrewise measures.

Oxtoby and Prasad (1978) proved that the Lebesque measure
$\lambda$ is the topologically unique exact (i.e. with full
support) and nonatomic (i.e. points have measure 0) probability
measure on the Hilbert cube $Q$.
We outline a proof of this theorem which can be parametrized.
Some consequences are the following:
Theorem 1. Let $\pi:Q\times Q\mapsto Q$ be the projection map onto
the second factor. Then $(\pi,\{ \lambda\times \delta_q \} _{q\in
Q})$ is a Milutin map with fibrewise measures $\{ \lambda\times
\delta _q \} $ (Here $ \delta
_q$ denotes Dirac measure). This
family of exact, atomless fiberwise measures associated with
$\pi$ is topologically unique.
Theorem 2. The above map $\pi$ with fibrewise measures
$\{\lambda\times \delta_q\}$ is a universal Milutin map.
Corollary. There is a continuous map $h : P_{\epsilon A}
(Q)\mapsto Homeo (Q)$ which assigns to each exact, atomless measure $\nu$ on $Q$ a homeomorphism $h(\nu)$ of
$Q$ which transforms the measure $\nu$ into the measure $\lambda$ i.e. $\nu(A)=\lambda(h(\nu)(A))$ for
each Borel set $A$ in $Q$.



16.10.06.

E.Tymchatyn


Continuous extension operators for uniformly continuous functions and pseudometrics.

Kunzi and Schapiro (1997) found a continuous linear extension operator for all realvalued continuous functions with domains which are compact subsets of a metric space (X, d). We find analogous theorems for the space of uniformly continuous bounded functions with domains which are bounded subsets of (X, d).



09.10.06.

Т.Banakh


Absolute Zspaces and their applications to
Dimension Theory.

A compact space X is defined an absolute Zspace if for any embedding of X into the Hilbert cube Q the set Xx{x_{0}} is a Zset in Qx[1,1]. We discuss the relation of absolute Zspaces to other dimension clases of compacta.



25.09.06, 2.10.06.

T.Radul


Asymptotic dimensions.

It is shown that the transfinite extension of the asymptotic counterpart of the large inductive dimension is not trivial.



11.09.06.

E.Tymchatyn, A.Zagorodnyuk


Free Banach spaces and extensions of Lipschitz maps.

We study the free Banach space B(X) over a metric space X, that is a predual space of the Banach space of all Lipschitz functions on X which preserve a marked point θ in X. Some applications to the extension theory of Lipschitz function are obtained.



02.09.06.

D.Repovs


Suspensions of celllike compactum.

We prove that
(1) Every compact metrizable space is weakly homotopy equivalent to a celllike compactum and
(2) There exists a noncontractible celllike compactum whose suspension is contractible (this
gives an affirmative answer to the BestvinaEdwards problem).

