

19.05.08

L.Karchevska


The functor of nonexpanding functionals is not open

The functor E of nonexpanding functionals was introduced by A.Stan'ko and J.Camargo.
It contains many known functors: hyperspace exp, space of probability measures P, superextension λ, space of inclusion hyperspaces G,
the functor of orderpreserving functionals O and many other as subfunctors.
All the functors mentioned above are open and hence bicommutative.
Theorem. The functor E is finitely open but fails to be open (moreover, E is not bicommutative).
A functor F is called (finitely) open if for any surjective open map f:X→ Y between (finite) compact spaces the map Ff: FX→ FY is open.



12.05.08

R.Voytsitskyy


Extensor and infinitedimensional properties of hyperspaces

Principal results of the Ph.D. Thesis of R.Voytsitskyy will be presented.



5.05.08

V.Mykhaylyuk


Some problems of the Baire and Lebesgue classification of separately continuous functions

The principal results of the Doctor Dissertation of V.Mykhaylyuk will be presented.



22.04.08

M.Zarichnyi
 


31.03.08

T.Banakh


Exdending vectorvalued functions

We prove that a Banach space Y is reflexive if and only if for each closed subspace A of a generalized ordered space X
there is a linear extender
u:C_{∞}(A,Y)→C_{∞}(X,Y) if and only if such an extender exists
for the subset of rationals on the Michael line. For the proof we introduce a relative version of the strong Choquet
game created by G.Choquet for characterizing Polish spaces.
This is a joint work with I.Banakh and Kaori Yamazaki.



24.03.08

T.Banakh


The coarse classification of countable groups

Two theorems are proved:
Theorem 1. Each countable group G of asymptotic dimension asdim(G)=0 is coarsely equivalent to the antiCantor set 2^{<ω}.
Theorem 2. A countable Abelian group G with asdim(G)=n is coarsely equivalent to:
 Z^{n} iff G is finitely generated or n=∞;
 Z^{n}×2^{<ω} iff G is infinitely generated.
This is a joint work with Jose Higes and Ihor Zarichnyi.



03.03.08,
17.03.08

T.Banakh


The coarse classification of homogeneous ultrametric spaces

We prove that two homogeneous ultrametric spaces are coarsely equivalent if and only
if they have the same sharp entropies.
This classification implies that each homogeneous proper ultrametric space is coarsely equivalent to the
antiCantor set. In particular, any two countable locally finite groups are coarsely equivalent.
For the proof of these results we develop a technique of towers which can have an independent interest.
This is a joint work with Ihor Zarichnyi.



18.02.08,
25.02.08

T.Radul, T.Zarichnyi, B.Bokalo
 


24.12.07

N.Lyaskovska


Packing index of subsets in Polish groups

For a subset A of a group G we study the packing index
ind_{P}(A)=sup{S:S⊂ G, {xA}_{x∈S} is disjoint} of A.
We show that the packing index of a σcompact subset of a Polish group cannot take an intermediate value between ℵ_{1}
and the contnuum c.
On the other hand, each nondiscrete Polish group contains a nowhere dense Haar null subset of arbitrary packing index κ≤c.



17.12.07

T.Banakh


Algebra in superextensions of groups

We extend the binary operation from a group G to a righttopological semigroup operation on the superextension λ(G) of G and study the
properties of the obtained supercompact righttopological semigroup. We prove that all minimal left ideals of the superextension λ(Z) are metrizable topological semigroups, isomorphic to
minimal left ideals of the superextension λ(Z_{2}) of the compact group Z_{2} of integer 2adic numbers.



10.12.07

O.Gutik


On finite partial bijections of bounded rank of a Hausdorff topological space.

We prove that the semigroup of finite partial bijections of bounded rank of a Hausdorff topological space is algebraically closed in the class of topological inverse semigroups.



3.12.07

L.Zdomskyy


The strong Pytkeev property in function spaces C_{k}(X) and C_{p}(X)

We prove that for each Polish space X, the space C(X) of continuous
realvalued functions on X satisfies (a strong version of) the Pytkeev
property, if endowed with the compactopen topology.
We also consider the Pytkeev property in the case where
C(X) is endowed with the topology of pointwise convergence.
This is a joint work with Boaz Tsaban.



26.11.07

L.Zdomskyy


The failure of the Lindelöf property in powers of Lspaces.

We show that under PFA a hereditarily Lindelöf space X is hereditarily separable provided all finite powers of X are Lindelöf.
This is a joint work with Boaz Tsaban.



19.11.07

L.Zdomskyy


Characterizing the Arkhangelski's α_{1}property in C_{p}spaces.

We show that for a perfectly normal space X with Ind X=0 the following conditions are equivalent:
(i) the function space C_{p}(X) has the Arkhangelski's property α_{1};
(ii) for every metrizable space Y the space B_{p}(X,Y) of all Borel maps from X to Y, endowed with the topology of pointwise
convergence, has the property α_{1};
(iii) the function space C_{p}(X,{0,1}) has the property α_{1.5} (which is formally weaker than α_{1});
(iv) for each Borel map f : X→ ω^{ω} the image f(X) lies in a σcompact subset of ω^{ω};
(v) the space X satisfies the Selection Principle U_{fin}(B,Γ).
This is a joint work with Boaz Tsaban.



12.11.07

T.Banakh


Righttopological semigroups of maximal linked systems on groups

We extend the binary operation from a group G to a righttopological semigroup operation on the superextension λ(G) of G and study the
properties of the obtained supercompact righttopological semigroup. We discuss possible applications of the semigroup λ(Z) to the
Problem of Owings who asked if for any partition of N into two pieces one of the pieces contains the sum set I+I of some infinite subset I.



5.11.07

D.Repovs


Characterization of smooth manifolds by smooth homogeneity

The talk will be devoted to a generalization of a problem originally
due to V. I. Arnol'd,
to arbitrary C^{∞}homogeneous compacta.
A locally compact subset
K of R^{n} is said to be C^{∞}homogeneous if for every
pair of points x,y in K there exist neighborhoods
O_{x}, O_{y} of x and y in R^{n}
and a C^{∞}diffeomorphism
h : (O_{x}, O_{x} ∩ K, x)→ (O_{y}, O_{y}∩ K, y). The following is a
characterization of
C^{∞}homogeneous subsets of the Euclidean space R^{n}:
Let K be a locally compact (possibly nonclosed) subset of R^{n}.
Then K is C^{∞}homogeneous if and only if K is a
C^{∞}submanifold of
R^{n}, i.e. (i) if dim K = 0, then K is at most countable subset
of
isolated points in R^{n}; (ii) if 0 < dim K < n, then K is at most
countable
collection of C^{∞}submanifolds
with pairwise disjoint neighborhoods in R^{n}; and
(iii) if dim K = n, then K is an open subset of R^{n}.
Our tools involve, among others, classical dimension theory (covering,
Hausdorff and cohomological dimension) and the Baire Category
Theorem. We shall also discuss further developments, e.g. the failure of such
a characterization for Lipschitz homogeneity, the connection with the
HilbertSmith Conjecture, related work in chaos theory, etc.



15.10.07, 22.10.07

T.Banakh


Each G_{δ}measurable map from a complete metric space into a regular space is F_{σ}measurable.

Answering a question of B.Bokalo, we prove that each G_{δ}measurable map from a complete metric space into a regular space
is F_{σ}measurable. A map f is called G_{δ}measurable (resp. F_{σ}measurable)
if the preimage of each open set is of type G_{δ} (resp. F_{σ}) in X. Details of the proof can be found in the paper
[T.Banakh, B.Bokalo, On scatteredly continuous maps between topological spaces.]



8.10.07.

T.Radul


Geometry of multiplication map of the orderpreserving monad.

We consider some properties of multiplication map of the monad of oderpreserving functionals and prove that it is soft for any compactum.



1.10.07.

T.Banakh


Homeomorphism groups of noncompact surfaces

We survey existing results on the topological structure of the homeomorphism groups of compact and noncompact surfaces.



17.09.07.

T.Banakh, B.Bokalo, O.Gutik


Divertissement.

Some open problems have been posed. In particular, B.Bokalo has asked if
each G_{δ}measurable map f from a Polish space X to a regular space Y is F_{σ}measurable.



10.09.07.

M.Zarichnyi, I.Guran, T.Radul


Divertissement.

Some open problems have been posed.



31.08.07.

D.Repovs


Interesting constructions based on the Topologist sine curve.

We shall present a variety of interesting applications of the classical example of a 1dimensional connected nonPeano planar continuum, the Topologist sine curve (and its derivatives, most notably the Warsaw circle) to diverse problems of geometric topology in dimensions 2 and 3. For example: an example showing that the classical van Kampen theorem fails without the openess condition, a counterexample to Molnar's theorem from 1950's, and a construction of a 2dimensional noncontractible simply connected celllike continua. We shall also present the solution of the BestvinaEdwards problem: Does there exist a noncontractible celllike compactum whose suspension is contractible? In conclusion we shall state some interesting open problems.

