We construct Bernstein sets in ℝ having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.

Some existence results on best approximation are proved without starshaped subset and affine mapping in the set up of $q$-normed space. First, we consider the closed subset and then weakly compact subsets for said purpose. Our results improve the result of Mukherjee and Som (Mukherjee, R. N., Som, T., A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16(3) (1985), 243–244.) and Jungck and Sessa (Jungck, G., Sessa, S., Fixed point theorems in best...

In this work we present results on fixed points, pairs of coincidence points and best approximation for ε-semicontinuous mappings in metric trees. It is a generalization of the similar properties of upper and almost lower semicontinuous mappings.

In this paper we study simultaneous approximation of $n$ real-valued functions in $L_p\left[a,b\right]$ and give a generalization of some related results.

We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega }_1^{CK}$, if for some $\eta <{\aleph }_1$ and all u ∈ WO of length η, a is ${\Sigma }_{\alpha }^0\left(u\right)$, then a is ${\Sigma }_{\alpha }^0$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma }_1^1$-Turing-determinacy implies the existence of $0^{}$.

We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures.

We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number 2 and Baire class ξ chromatic number ℵ₀.

We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $\Sigma {⁰}_{\alpha }$, α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $\Pi {⁰}_{\alpha }$ uniformization which is the graph of a $\Sigma {⁰}_{\alpha }$-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{\delta }$ sections.

We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A\left(ℰ\otimes ℬ\right(X\left)\right)$ into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations...

For a linear operator T in a Banach space let ${\sigma }_p\left(T\right)$ denote the point spectrum of T, let ${\sigma }_{p,n}\left(T\right)$ for finite n > 0 be the set of all $\lambda \in {\sigma }_p\left(T\right)$ such that dim ker(T - λ) = n and let ${\sigma }_{p,\infty }\left(T\right)$ be the set of all $\lambda \in {\sigma }_p\left(T\right)$ for which ker(T - λ) is infinite-dimensional. It is shown that ${\sigma }_p\left(T\right)$ is ${ℱ}_{\sigma }$, ${\sigma }_{p,\infty }\left(T\right)$ is ${ℱ}_{\sigma \delta }$ and for each finite n the set ${\sigma }_{p,n}\left(T\right)$ is the intersection of an ${ℱ}_{\sigma }$ set and a ${}_{\delta }$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more detailed decomposition...