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Let X be a nonempty set of cardinality at most $2^{\aleph ₀}$ 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.

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

Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.

We consider the following notion of largeness for subgroups of $S_{\infty }$. 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 $S_{\infty }$ can be extended to a large free subgroup of $S_{\infty }$, and, under Martin’s Axiom, any free subgroup of $S_{\infty }$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty }$. Finally, if Gₙ are countable groups, then...

Jachymski showed that the set $$\left\{(x,y)\in {𝐜}_0\times {𝐜}_0:{\left(\sum _{i=1}^n\alpha \left(i\right)x\left(i\right)y\left(i\right)\right)}_{n=1}^{\infty }\text{is}\text{bounded}\right\}$$ is either a meager subset of ${𝐜}_0\times {𝐜}_0$ or is equal to ${𝐜}_0\times {𝐜}_0$. In the paper we generalize this result by considering more general spaces than ${𝐜}_0$, namely ${𝐂}_0\left(X\right)$, the space of all continuous functions which vanish at infinity, and ${𝐂}_b\left(X\right)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.

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 $E=\{(f_1,...,f_n)\in L^{p_{1,}{q}_1}\times \cdots \times L^{p_n,q_n}:f_1\cdots f_n\in L^{p,q}\}$. We prove the following dichotomy: either $E=L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$ or E is σ-porous in $L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E=L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$ or E is meager. This is a generalization of the results for classical L p spaces.

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ${\sum }_{n=1}^{\infty }x\left(n\right)$. 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 ${\sum }_{n=1}^{\infty }b\left(n\right)$ 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...

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 $L^q\left(U\right)$. 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,...

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 $\sigma$-ideal, being (completely) nonmeasurable with respect to different $\sigma$-ideals, being a $\kappa$-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...