Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2006/07) academic year



Topology
&
Applications

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 compact-valued 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 3-dimensional space R3 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 X3 such that H(x,y,z)=(x,y,hx,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-1z). An example of a continuously homogeneous space homeomorphic to no topological group is the 7-dimensional sphere S7. 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
  • Group-sequential topological groups.
    • A topological group G is called group-sequential if it carries the strongest group topology inducing the original topology on each convergent sequence. We give examples of such topological groups, characterize group-sequential free topological groups and pose some open problems.
12.03.07. O.Gutik
  • H-closed topological semigroups and semilattices.
    • In the report we shall discuss about some properties of H-closed topological semigroups and semilattices.
05.03.07. K.Koporkh
  • On spaces of quotient maps.
26.02.07. M.Zarichnyi
  • Spaces of upper-semicontinuous capacities.
    • The notion of capacity was introduced into mathematics by G.Choquet. The family of all upper-semicontinuous 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 Hartman-Mycielski functor.
    • We investigate topological properties of Hartman-Mycielski 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 $f|A$ 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 sigma-ideals.
15.11.06. Alexander Balinsky
  • Zero modes of Pauli operator
13.11.06. P. Borodulin-Nadzieja
  • 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 non-compact 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 Ui with Ui+1 refining Ui, 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 non-atomic (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 real-valued 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 Z-spaces and their applications to Dimension Theory.
    • A compact space X is defined an absolute Z-space if for any embedding of X into the Hilbert cube Q the set Xx{x0} is a Z-set in Qx[-1,1]. We discuss the relation of absolute Z-spaces 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 cell-like compactum.
    • We prove that
      (1) Every compact metrizable space is weakly homotopy equivalent to a cell-like compactum and
      (2) There exists a noncontractible cell-like compactum whose suspension is contractible (this gives an affirmative answer to the Bestvina-Edwards problem).