Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2007/08) academic year



Topology
&
Applications

19.05.08 L.Karchevska
  • The functor of non-expanding functionals is not open
    • The functor E of non-expanding 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 order-preserving 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 infinite-dimensional 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 vector-valued 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 anti-Cantor set 2.
      Theorem 2. A countable Abelian group G with asdim(G)=n is coarsely equivalent to:
      - Zn iff G is finitely generated or n=∞;
      - Zn×2 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 ultra-metric spaces
    • We prove that two homogeneous ultra-metric spaces are coarsely equivalent if and only if they have the same sharp entropies. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the anti-Cantor 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
  • Divertissement
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 indP(A)=sup{|S|:S⊂ G,   {xA}x∈S is disjoint} of A. We show that the packing index of a σ-compact subset of a Polish group cannot take an intermediate value between ℵ1 and the contnuum c. On the other hand, each non-discrete 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 right-topological semigroup operation on the superextension λ(G) of G and study the properties of the obtained supercompact right-topological 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(Z2) of the compact group Z2 of integer 2-adic 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 Ck(X) and Cp(X)
    • We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies (a strong version of) the Pytkeev property, if endowed with the compact-open 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 L-spaces.
    • 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 α1-property in Cp-spaces.
    • We show that for a perfectly normal space X with Ind X=0 the following conditions are equivalent: (i) the function space Cp(X) has the Arkhangelski's property α1; (ii) for every metrizable space Y the space Bp(X,Y) of all Borel maps from X to Y, endowed with the topology of pointwise convergence, has the property α1; (iii) the function space Cp(X,{0,1}) has the property α1.5 (which is formally weaker than α1); (iv) for each Borel map f : X→ ωω the image f(X) lies in a σ-compact subset of ωω; (v) the space X satisfies the Selection Principle Ufin(B,Γ).
      This is a joint work with Boaz Tsaban.
12.11.07 T.Banakh
  • Right-topological semigroups of maximal linked systems on groups
    • We extend the binary operation from a group G to a right-topological semigroup operation on the superextension λ(G) of G and study the properties of the obtained supercompact right-topological 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 Rn is said to be C-homogeneous if for every pair of points x,y in K there exist neighborhoods Ox, Oy of x and y in Rn and a C-diffeomorphism h : (Ox, Ox ∩ K, x)→ (Oy, Oy∩ K, y). The following is a characterization of C-homogeneous subsets of the Euclidean space Rn: Let K be a locally compact (possibly nonclosed) subset of Rn. Then K is C-homogeneous if and only if K is a C-submanifold of Rn, i.e. (i) if dim K = 0, then K is at most countable subset of isolated points in Rn; (ii) if 0 < dim K < n, then K is at most countable collection of C-submanifolds with pairwise disjoint neighborhoods in Rn; and (iii) if dim K = n, then K is an open subset of Rn. 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 Hilbert-Smith Conjecture, related work in chaos theory, etc.
15.10.07,
22.10.07
T.Banakh
  • Each Gδ-measurable map from a complete metric space into a regular space is Fσ-measurable.
    • Answering a question of B.Bokalo, we prove that each Gδ-measurable map from a complete metric space into a regular space is Fσ-measurable. A map f is called Gδ-measurable (resp. Fσ-measurable) if the preimage of each open set is of type Gδ (resp. Fσ) 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 order-preserving monad.
    • We consider some properties of multiplication map of the monad of oder-preserving functionals and prove that it is soft for any compactum.
1.10.07. T.Banakh
  • Homeomorphism groups of non-compact surfaces
    • We survey existing results on the topological structure of the homeomorphism groups of compact and non-compact 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 Gδ-measurable map f from a Polish space X to a regular space Y is Fσ-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 1-dimensional connected non-Peano 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 2-dimensional noncontractible simply connected cell-like continua. We shall also present the solution of the Bestvina-Edwards problem: Does there exist a noncontractible cell-like compactum whose suspension is contractible? In conclusion we shall state some interesting open problems.