Let K be a bounded, closed and convex subset of a Banach space X. We show that the iterates of a typical element (in the sense of Baire category) of a class of nonexpansive mappings which take K to X converge uniformly on K to the unique fixed point of this typical element.

There is a well-known global equivalence between Σ2 sets having the Universal Baire property, two-step Σ3 generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of Σ2 sets being (2 )-cc Universally Baire, which is below 0. In a model obtained, there is a Σ2 set which is weakly ω2-Universally Baire but not ω2-Uni...

In [4], we introduced determinacy schemata motivated by Wadge classes in descriptive set theory. In this paper, we prove that a simple iteration of Σ1 inductive definition implies Sep(∆ 0 2,Σ 0 2) determinacy in the Baire space over RCA0.

We investigate the subset E of Teichmuller space consisting of all surfaces which have at least two closed simple geodesics of the same length. Using elementary methods from the theory of surfaces we show that E is baire meagre but its complement, N , contains no arcs.

We exhibit a class of nonlinear operators with the property that their iterates converge to their unique fixed points even when computational errors are present. We also show that most (in the sense of the Baire category) elements in an appropriate complete metric space of operators do, in fact, possess this property.

We consider continuous descent methods for the minimization of convex functions defined on a general Banach space. In our previous work we showed that most of them (in the sense of Baire category) converged. In the present paper we show that convergent continuous descent methods are stable under small perturbations.

We study the recently suggested effective Wadge hierarchy in spaces, concentrating on non-collapse property. Along with hierarchies of sets, we k-partitions which are interesting their own. In particular, establish sufficient conditions for and apply them to some concrete including discrete space natural numbers, Baire space, Cantor space. While latter two cases obtained by easy effectivization...

A point in Baire space is found for which the first derived ω-limit set is not Borel, whilst the second is empty. A second point is found for which the sequence of derived ω-limit sets does not stabilise until the first uncountable ordinal. The two points are recursive.

§0. Introduction The Baire category theorem implies that the family, F , of dense sets Gδ in a fixed metric space, X, is a candidate for generic sets since it is closed under countable intersections; and if X is perfect (has no isolated point), then A ∈ F has uncountable intersections with any open ball in X. There is a long tradition of soft arguments to prove that certain surprising sets are ...

We say that E ⊆ X × X is a clopen graph on X iff E is symmetric and irreflexive and clopen relative to X\∆ where ∆ = {(x, x) : x ∈ X} is the diagonal. Equivalently E ⊆ [X] and for all x 6= y ∈ X there are open neighborhoods x ∈ U and y ∈ V such that either U × V ⊆ E or U × V ⊆ X\E. For clopen graphs E1, E2 on spaces X1, X2, we say that E1 continuously reduces to E2 iff there is a continuous map...