Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2013/14) academic year



Topology
&
Applications

Date: Speaker:       Title (click on the title to see the Abstract):
26.05.14 M.Pylypovych
M.Simkiv
  • Generalizing Kuratowski Theorem about 14 sets
    • Recently, Shalitt and Willard showed that Kuratowski 14-set theorem does not have an immediate generalization for multitopological spaces. Situation changes if the topologies are comparable. For a multitopological space with comparable topologies we prove the finiteness of number of possible sets obtained by operations of closure and complement. We also determine the maximal possible number of different sets obtained by these operations in a multitopological space and prove that this number is indeed achieved on some subset of a suitable n-topological space.
      The sequence 2,14,126,1394,... of these cardinalities is not included in Online Encyclopedia of Integer sequences.
      This is a joint result with T.Banakh, O.Chervak, T.Martynyuk, and A.Ravsky
19.05.14 T.Radul
  • Convex hulls of functional functors
    • We define and study the convexification of functional functors and discuss some related open problems.
05.05.14
12.05.14
T.Banakh
  • On difference sets in partitions of G-spaces and groups
    • Using idempotent quasi-invariant measures on groups we shall prove that for each partition G=A1∪...∪An of a group G there is a finite subset F of cardinality |F|≤n in G such that G=FAiAi-1Ai for some cell of the partition. This gives a partial answer to a problem of I.V.Protasov posed in 1995 in the Kourovka Problem Notebook.
7.04.14 M.Zarichnyi
  • On free Banach lattices
    • We consider some categorial properties of the construction of pairs of convex sets in Banach spaces and prove that it leads to the construction of a free Banach lattice over a Banach space.
31.03.14 T.Banakh
  • On convergent sequences in scattered compact semilattices
    • We shall prove that for any point x of countable scattered height in a compact topological semilattice S, any sequence (xn)n∈ω that converges to x and any monotone sequences (xn,m)m∈ω that converge to points xn there are a function f:ω→ω and an infinite subset A of ω such that the diagonal subsequence (xn,f(n))n∈A converges to x.
      Applying this theorem we give a simple proof of a difficult theorem of Banakh, Gutik and Rajagopalan who proved in 2005 that each point of scattered height ≤ 2 in a compact topological semilattice is the limit of a convergent sequence of isolated points of S.
24.03.14 T.Banakh
  • Canonical subsemilattices in scattered compact semilattices of low scattered height
    • We shall prove that each locally minimal point of scattered height 2 in a compact topological semilattice S is contained in a compact countable subsemilattice T that contain dense set of isolated points of S and is topologically isomorphic to one of 7 canonical semilattices.
      This is a joint work with R.Bonnet and W.Kubis.
03.03.14
17.03.14
T.Banakh
  • Attractors of contractive function systems on multimetric spaces
    • We prove counterparts of Banach and Matkowski Fixed Point Theorems for contractive self-maps of multimetric spaces and apply the obtained theorems to studying attractors of contractive function systems on topologicals spaces. In particular, we prove a metrization theorem for topologically contracting function systems.
24.02.14
All
17.02.14
T.Radul
  • Fixed Point and Nash equilibrium Theorems for binary convexities
    • We prove counterparts of Kakutani Fixed Point Theorem and Nash Equilibrium Theorem for binary convexities and apply the obtained results to binary monads. As a corollary we obtain the Kozhan-Zarichnyi Equilibrium Theorem for capacities.
10.02.14
All
  • Divertissement
    • Active participants of the seminar will present and discuss interesting open problems from various branches of topology and its applications.
3.02.14 I.Hetman
  • Topological structure of the hyperspaces of closed convex subsets in normed spaces
    • Some principal results the Ph.D. Thesis of Ivan Hetman is discussed.
* * *
  • * * * Winter Holidays * * *
    • * * *
23.12.13
30.12.13
I.Guran
  • Boundedness in topological rings
    • We shall discuss various notions of boundedness in topological rings. In particular, we prove that each compact topological ring is totally disconnected.
16.12.13
A.Ravsky
  • On semiregular semitopological groups, II
    • We consider the problem of continuity of the multiplication on groups, close to being compact.
9.12.13
T.Radul
  • Tensor products versus monads
    • It is well-known that each monadic functor admits a tenor product. We shall discuss the existence of tensor products for functors which cannot completed to a monad.
25.11.13
A.Ravsky
  • On semiregular semitopological groups
    • We consider the problem of continuity of the multiplication on groups, close to being compact.
18.11.13
M.Zarichnyi
  • The asymptotic property C
    • We shall discuss the asymptotic property C in the coarge category and detect some groups with this property.
11.11.13
B.Bokalo
  • On some classes of discontinuous functions
    • We discuss the relation between some classes spaces (scattered, fragmentable) and functions (scatteredly continuous, weakly discontinuous, feebly continuous, etc). In particular, we prove that a fragmentable compact Hausdorff space is metrizable if and only if it is hereditarily Lindelof.
4.11.13
T.Banakh
  • Pytkeev ℵ0-spaces
    • We introduce a new class of generalized metric spaces consisting of Pytkeev ℵ0-space, i.e., a regular topological space possessing a countable Pytkeev network. The class of Pytkeev ℵ0-spaces contains all metrizable separable spaces and is (properly) contained in the class of ℵ0-spaces. This class is closed under many operations over topological spaces: taking a subspace, a countable Tychonoff product, a countable small box-product, a direct limit topology, taking the hyperspace and a function space endowed with the compact-open topology. More precisely, for every Hausdorff ℵ0-space X and a Pytkeev ℵ0-space Y the function space Ck(X,Y) endowed with the compact-open topology is a Pytkeev ℵ0-space. This enforces a classical result of E.Michael (1966) on function spaces between ℵ0-spaces. Pytkeev ℵ0-spaces can be used to get a new characterization of second countable spaces: a topological space is second countable if and only if it is a Pytkeev ℵ0-space with countable fan tightness.
29.10.13
O.Chervak
  • Kuratowski 14-set problem for multitopological spaces.
    • Recently, Shalitt and Willard showed that Kuratowski 14-set theorem does not have an immediate generalization for multipological spaces. Situation changes if topologies are comparable. For a multitopological space with comparable topologies we prove the finiteness of number of possible sets obtained by operations of closure and complement. We also determine the maximal possible number of different sets obtained by these operation in a multitopological space.
      Sequence 2,14,126,1394, of these cardinalities is not included in Online Encyclopedia of Integer sequences.
      This is a joint result with T.Banakh, T.Martynyuk, M. Pylypovych, A.Ravsky
21.10.13
I.Pastukhova
  • Automatic continuity of homomorphisms between topological semigroups
    • In 1976 Yeager proved that a homomoprhism h:X→Y between compact topological Clifford semigroups is continuous if and only if for any subsemilattice E and any subgroup H of X the restrictions h|E and h|H are continuous. In this talk we shall discuss possible extensions of this Yeager's theorem beyond the class of compact topological Clifford semigroups.
      This is a joint result with T.Banakh.
30.09.13
T.Banakh
  • Metrizability of Lawson compact pre-Clifford semigroups
    • A semigroup S is called pre-Clifford if it is a union of groups. A topological pre-Clifford semigroup S is called Lawson if for each idempotent e of S and its neighborhood O(e) in S there is an open subsemigroup U of S which contains the maximal subgroup π-1(e) and is contained in the open set π-1(O(e)). Here π:S→ E, π:x→xx-1=x-1x, is the projection of S onto the set E of idempotents of S.
      It is proved that a Lawson pre-Clifford compact topological semigroup S is metrizable if and only if the set E of idempotents of S is metrizable and each maximal subgroup He-1(e), e in E, is metrizable.
      This is a joint result with I.Pastukhova and O.Potyatynyk.
28.09.13
All
  • Топологічна прогулянка
    • В суботу 28 вересня у парку "Знесіння" (біля озера) відбулася традиційна топологічна прогулянка активних учасників топологічного семінару (T.Банах, O.Равський, Т.Мартинюк) та гостей семінару (A.Bartoszewicz, Sz.Glab, F.Strobin, M.Bienias) з Politechniki Lodzkiej.




23.09.13
O.Gutik
  • Primitive inverse pseudocompact (semi)topological semigroups
    • The speaker presented some results on structure of primitive inverse pseudocompact (semi)topological semigroups.
16.09.13
O.Ravsky
  • On 2-pseudocompact paratopological groups which are topological groups
    • It is proved that a 2-pseudocompact Hausdorff paratopological group is a topological group if it contains a compact subset with countable pseudocharacter.
9.09.13
All
  • Divertissement
    • Active participants of the seminar will present and discuss interesting open problems from various branches of topology and its applications.