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On the Converse of Caristi's Fixed Point Theorem

100%

Głąb S.

nr 4

411-416

Let X be a nonempty set of cardinality at most $2^{ℵ₀}$ and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.

Large free subgroups of automorphism groups of ultrahomogeneous spaces

63%

Głąb S. , Strobin F.

nr 2

279-295

We consider the following notion of largeness for subgroups of $S_{∞}$. A group G is large if it contains a free subgroup on 𝔠 generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{∞}$ can be extended to a large free subgroup of $S_{∞}$, and, under Martin's Axiom, any free subgroup of $S_{∞}$ of cardinality less than 𝔠 can also be extended to a large free subgroup of $S_{∞}$. Finally, if Gₙ are countable groups, then either $∏_{n∈ℕ} Gₙ$ is large, or it does not contain any free subgroup on uncountably many generators.

Descriptive properties of density preserving autohomeomorphisms of the unit interval

63%

Głąb S. , Strobin F.

tom 8

nr 5

928-936

We prove that density preserving homeomorphisms form a Π11-complete subset in the Polish space ℍ of all increasing autohomeomorphisms of unit interval.

Large structures made of nowhere $L^{q}$ functions

51%

Głąb S. , Kaufmann P. , Pellegrini L.

nr 1

13-34

We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction $f|_U$ is not in $L^{q}(U)$. When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p's but nowhere q-integrable for some other q's (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González, improve and complement recent spaceability and algebrability results of several authors and motivate new research directions in the field of spaceability.

Algebraic and topological properties of some sets in ℓ₁

51%

Banakh T. , Bartoszewicz A. , Głąb S. , Szymonik E.

nr 1

75-85

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $∑_{n=1}^{∞} x(n)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: (𝓘) a finite union of closed intervals; (𝓒) homeomorphic to the Cantor set; 𝓜 𝓒 homeomorphic to the set T of subsums of $∑_{n=1}^{∞} b(n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, 𝓒 and 𝓜 𝓒 the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), (𝓒) and (𝓜 𝓒), respectively. We show that ℐ and 𝓒 are strongly 𝔠-algebrable and 𝓜 𝓒 is 𝔠-lineable. We also show that 𝓒 is a dense $𝒢_δ$-set in ℓ₁ and ℐ is a true $ℱ_σ$-set. Finally we show that ℐ is spaceable while 𝓒 is not.