We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a de-finable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. The converse is clearly not true. However,...

Introduction. It is well known that a function which is a derivative is in the first Baire class, and it is also known that a function which is an approximate derivative is in the first Baire class. The analogous statement for partial derivatives of functions of more than one variable is of course not true. For if g is a nonmeasurable function of one variable, then the function f(x, y)=x-g(y) h...

We shall exposit here one of the basic theorems leading from large cardinals to determinacy, a result of Woodin known as the derived model theorem. The theorem dates from the mid80’s and has been exposited in several sets of informally circulated lecture notes (e.g., [12], [14]), but we know of no exposition in print. We shall also include a number of subsidiary and related results due to vario...

All polynomial many-one degrees are shown to be of second Baire category in the superset topology when witness functions are allowed to run in 2 log h n time, for any h. Any improvement of this result for the complete p-m-degrees of RE, EXP or NP implies P 6 = NP.

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.

In this paper we study the conditions under which a fuzzy topological space becomes a fuzzy almost resolvable space and the inter-relations between fuzzy almost resolvable, fuzzy almost irresolvable spaces, fuzzy submaximal spaces, fuzzy first category spaces, fuzzy Baire spaces, fuzzy weakly Volterra spaces are also investigated.

We study some set-theoretic properties of Schmidt’s σ-ideal on R, emphasizing its analogies and dissimilarities with both the classical σ-ideals on R of Lebesgue measure zero sets and of Baire first category sets. We highlight the strict analogy between Schmidt’s ideal on R and Mycielski’s ideal on 2N.

We investigate combinatorial issues relating to the use of random orbit approximations to the attractor of an iterated function system with the aim of clarifying the role of the stochastic process during generation the orbit. A Baire category counterpart of almost sure convergence is presented; and a link between topological and probabilistic methods is observed.