Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2012/13) academic year


Date: Speaker:       Title (click on the title to see the Abstract):
  • Metrization of the functor of idempotent measures with applications to idempotent fractals
    • Answering a question of M.Zarichnyi we find a metrization of the functor of idempotent measures. For this we construct an embedding of the space I(X) of idempotent measures on a compact Hausdorff space X to the hyperspace exp(I× X) and endow I(X) with the metric induced from the Hausdorff metric on I×X. Using this metrization, we prove that any contracting maps f1,...,fn: X→X on a metric compact space X and any non-negative real numbers p1,...,pn with max{p1,...,pn} = 1 the map Φ = max1≤i≤npiI(fi): I(X)→I(X) has a unique fixed point, called the idempotent fractal, which can be found as the limit of the sequence of iterations Φn0), n=1,2,3,...
  • On H-closed pospaces
    • We give a criterion for H-closedness of a point separated pospace with compact maximal antichains.
  • Evaluating the cardinality of Kuratowski monoid of an n-topological space
    • We prove that the sequence K(n)=∑ni,j=0(Cii+j)2 equal to the cardinality of the free Kuratowski of type (n,n) has asymptotic growth (1+o(1))16n+1/9πn as n→∞.
  • Kuratowski operations in n-topological spaces
    • Given a set X endowed with n comparable topologies we calculate the number of sets that can be obtained from a given set A of X by repeated application of the operations of closure and interior in these topologies.
  • Boundedness and separability in topological groups, II
    • We shall introduce several (more precisely, infinitely many) cardinal characteristics expressing boundedness and separability properties of topological groups and shall present several distinguishing examples. Also we shall calculate these cardinal characteristics for the symmetric group S(X) and its dense normal subgroup Sω(X) consisting of finitely supported permutations of an infinite set X.
  • Limits of bounded degree graphs and hyperfiniteness.
    • This is an expository talk based on an article of G.Elek. We introduce notions of Benjamini-Schramm limit and graphing, give examples. We define a notion of hyperfiniteness for graphings and graph sequences and prove that graph sequence is hyperfinite iff its limit graphing is hyperfinite. We mention the Kaimanovich theorem and prove that weakly equivalent graphings are hyperfinite simultaneously. We briefly discuss different notions of equivalent graphings (weakly, local-global, strong) and mention that they coincide for hyperfinite ones.
  • Boundedness and separability in topological groups
    • A topological group G is called
      - ω-bounded if for each neighborhood U of the unit 1G in G there is a countable set C in G such that CU=G=UC;
      - separable if there is a countable set C in G such that CU=G=UC for each neighborhood U of the unit 1G in G;
      - ω-duobounded if for each neighborhood U of the unit 1G in G there is a countable set C in G such that G=CUC;
      - duoseparable if there exists a countable set C in G such that G=CUC for each neighborhood U of the unit 1G in G.
      It is clear that each separable topological group is ω-bounded and duoseparable. On the other hand, a topological group is ω-doubounded if it is duoseparable or ω-bounded. We show that for any uncountable set X the group FSym(X) of finitely supported permutations of X endowed with the topology of pointwise convergence is ω-duobounded but not ω-bounded and not duoseparable. Also we prove that each topological group embeds into a duoseparable (and hence ω-duobounded) topological group. This answers a question of I.Guran.
  • On invariant partitions of groups
    • We prove that for each partition G=A1∪...∪An of an infinite group G there is a cell Ai of the partition and a subset F⊂G of cardinality |F|≤n such that G=F (AiAi-1)G where AG=∪x∈G xAx-1 for a subset A⊂G. This implies that for any partition G=A1∪...∪An of G into inner-invariant sets Ai=AiG there is a cell Ai of the partition such that G=FAiAi-1 for some finite subset F⊂G of cardinality |F|≤n.
      This is a joint work with I.V.Protasov and S.Slobodianiuk.
  • Hyperspaces of max-plus convex subsets in the function spaces
    • The notion of max-plus convex set can be naturally defined for the spaces of continuous functions. The main result is the following one: the hyperspace of compact max-plus convex sets in the Banach lattice C(X) of continuous functions on an infinite compact space X is homeomorphic to the separable Hilbert space.
      This is a joint work with Dusan Repovs.
  • Absolute retracts and equiconnected monads
    • We introduce a general notion of equiconnected functor and show that each such functor has similar topological properties as probability measure functor and idempotent measure functor, which have some natural equiconnectedness structures.
  • Nonstandard Methods in Topology and Number Theory
    • Classification of Mahler of complex numbers and local distances (by P.Philippon). Diophantian approximation groups and nonstandard methods (by T.Gendron).
  • On Slobodianiuk's approach to solution of a Protasov's problem about partitions of G-spaces
    • We shall discuss recent progress in resolving a Protasovs' problem on partitions of G-spaces by Slobodianiuk's method.
  • Divertissement
    • Participants of the seminar present some problems they have recently investigated.
  • W i n t e r     H o l y d a y s
  • Pseudo-random permutations
    • A sequence of permutations is called pseudo-random if it contains each permutation of length k with frequency 1/k!+o(1). Applying the analytic language of limits of permutations we shall show that it suffices to check the case k = 4.
  • Small subset property of metric spaces.
    • In the spirit of Menger-Urysohn theorem for meager subsets in manifolds we define a notion of a small subset property (ssp) for metric spaces. Spaces with ssp are coarse analogues of topological manifolds. We prove that n-dimensional hyperbolic an euclidean spaces have ssp. We discuss properties of groups with ssp.
  • Max-min measures on ultrametric spaces
    • We construct a natural ultrametric on the set of max-min measures on an ultrametric space and establish some functorial properties of the corresponding constructions.
  • The language of category theoretic diagram chasing for set theory and topology
    • In the first part of the talk I shall describe the formalism of a labelled category able to describe some aspects of basic notions in set theory (equicardinality, subset, cofinality) and demonstrate how some diagram chasing constructions correspond to set theory arguments. This formalism satisfies the axioms of a Quillen model category, a formalism for homotopy theory; and thus indicates a connection between set theory and homotopy theory.
T. Banakh
  • Subamenable groups and their partitions
    • A group G is called subamenable if it admits a left-invariant monotone subadditive function μ defined on the family of all subsets of G such that μ≠1 and for each subset A of G of measure μ(A)<1 and each ε>0 there is a left large subset L of X\A of measure μ(L)<ε. We prove that the class of subamenable groups includes (i) all groups that has subamenable quotient group, (ii) all infinite groups admitting a totally bounded group topology, (iii) all groups containing an infinite amenable subgroup. We prove that for every natural k a countable subamenable group G admits a partition G=A∪ B such that for every k-element subset K of G the sets KA and KB are not left thick. This resolves one problem of I.Protasov.
      T.Banakh, I.V.Protasov, S.Slobodianiuk, Subamenable groups and their partitions, preprint.
T. Banakh
  • Some examples of compact topological inverse semigroups
    • We present some canonical examples of compact topological inverse semigroups and discuss their role in the structure theory of compact topological inverse semigroups.
T. Banakh
  • The Solecki submeasure on a group
    • The Solecki submeasure σ on a group G assigns to each subset A of G the real number σ(A)=infFmaxx,y|F∩ xAy|/|F| where the infimum is taken over all non-empty finite subsets of G. We shall discuss properties of the Solecki submeasure and its interplay with the Haar measure λ on a compact topological group G. In particular we show that σ(K)=λ(K) for each closed subset K of G.

      1. T. Banakh, The Solecki submeasure on a group, preprint.
29.10.12 T. Radul
  • Binary convexities and monads
    • The notion of L-monads was introduced in [1]. They have a functional representation, which preserve the monad structure. In [2] it was introduced a convexity structure on each F-algebra for any L-monad F in the category of compact Hausdorff spaces and continuous maps. This general construction of convexities includes known convexities for probability measures, superextensions, inclusion hyperspaces, etc. It was proved in [2] that each binary monad (i.e. a monad generating a binary convexity) has good topological properties. In particular, its functorial part transforms openly generated compacta into Dugundji compacta. We will prove that the latter property characterizes binary monads.

      1. T. Radul, On functional representations of Lawson monads, Applied Categorical Structures 9 (2001), 457–463.
      2. T. Radul, Convexities generated by monads, Applied Categorical Structures, 19 (2011), 729–739.
22.10.12 O.Chervak
  • An introduction to forcing
    • We give a short introduction to forcing technics, briefly explain general ways of obtaining various independency results. Also we build a model where Martin axiom holds and the Higson corona of the Cantor macro cube is homeomorphic to the Stone-Cech compactification of integers.
15.10.12 T. Radul
  • The Hartman-Mycielski functor cannot be completed to a monad
    • We prove that the Hartman-Mycielski functor cannot be completed to a (weak) monad in the category of compacta.
8.10.12 O. Chervak
  • Properties of Higson corona under different set-theoretic assumptions
    • Higson corona is a natural object in coarse geometry, being a coarse analogue of Stone-Cech compactification. As a consequence to Parovicenko theorems, I. Protasov proved that under CH Higson coronas of all asymptotically zero-dimensional metric spaces are homeomorphic and wondered if this result remains true in ZFC. We answer negatively on this question and prove that under some set-theoretic assumptions Higson corona of anti-Cantor set is not homeomorphic to Higson corona of divergent sequence
1.10.12 R.Cauty
  • Singular chains and fixed points of maps with a compact iterate
    • We shall explain some generic ideas underlying the use of singular chains for the proof of the following Fixed Point Theorem answering an old problem of Juliusz Shauder:
      Theorem. A continuous map f:C→C defined on a convex subset C of a linear metric space has a fixed point if for some iterate fn of f the image fn(C) is contained is a compact subset of C.
24.09.12 R.Cauty
  • A solution of West's Problem ANR 13
    • We shall prove that a metrizable σ-compact space X is an ANR if and only if X is an ANE(Comp). The latter means that each continuous map f:A→X defined on a closed subspace A of a compact metrizable space B admits a continuous extension F:O(A)→X defined on a neighborhood O(A) of A in B. This resolves a long-standing Problem ANR 13 from the West's list of open problems included to the book "Open Problems in Topology" (1990).
10.09.12 All
  • Divertissement
    • The active participants of the topological seminar will discuss new (summer) results and pose some open problems.