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On the Converse of Caristi's Fixed Point Theorem

Szymon Głąb — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

Let X be a nonempty set of cardinality at most 2 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

Szymon GłąbFilip Strobin — 2015

Colloquium Mathematicae

We consider the following notion of largeness for subgroups of S . A group G is large if it contains a free subgroup on 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 can also be extended to a large free subgroup of S . Finally, if Gₙ are countable groups, then...

Dichotomies for 𝐂 0 ( X ) and 𝐂 b ( X ) spaces

Szymon GłąbFilip Strobin — 2013

Czechoslovak Mathematical Journal

Jachymski showed that the set ( x , y ) 𝐜 0 × 𝐜 0 : i = 1 n α ( i ) x ( i ) y ( i ) n = 1 is bounded is either a meager subset of 𝐜 0 × 𝐜 0 or is equal to 𝐜 0 × 𝐜 0 . In the paper we generalize this result by considering more general spaces than 𝐜 0 , namely 𝐂 0 ( X ) , the space of all continuous functions which vanish at infinity, and 𝐂 b ( X ) , the space of all continuous bounded functions. Moreover, we replace the meagerness by σ -porosity.

Dichotomies for Lorentz spaces

Szymon GłąbFilip StrobinChan Yang — 2013

Open Mathematics

Assume that L p,q, L p 1 , q 1 , . . . , L p n , q n are Lorentz spaces. This article studies the question: what is the size of the set E = { ( f 1 , . . . , f n ) L p 1 , q 1 × × L p n , q n : f 1 f n L p , q } . We prove the following dichotomy: either E = L p 1 , q 1 × × L p n , q n or E is σ-porous in L p 1 , q 1 × × L p n , q n , provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either E = L p 1 , q 1 × × L p n , q n or E is meager. This is a generalization of the results for classical L p spaces.

Algebraic and topological properties of some sets in ℓ₁

Taras BanakhArtur BartoszewiczSzymon GłąbEmilia Szymonik — 2012

Colloquium Mathematicae

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: () a finite union of closed intervals; () homeomorphic to the Cantor set; 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, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable. We...

Large structures made of nowhere L q functions

Szymon GłąbPedro L. KaufmannLeonardo Pellegrini — 2014

Studia Mathematica

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,...

Two point sets with additional properties

Marek BieniasSzymon GłąbRobert RałowskiSzymon Żeberski — 2013

Czechoslovak Mathematical Journal

A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some σ -ideal, being (completely) nonmeasurable with respect to different σ -ideals, being a κ -covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations...

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