The following two theorems give the flavour of what will be proved. THEOREM. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0, 1] to Y coincide if and only if Y is connected and locally connected. THEOREM. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class fun...

This paper introduces a new deenition of resource-bounded Baire category in the style of Lutz. This deenition gives an almost-all/almost-none theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new deenition than in previous deenitions. For example, almost no sets in EXP are EXP-complete, and NP is PF-meager unless NP = EXP. It is also s...

Working within weak subsystems of second-order arithmetic Z2 we consider two versions of the Baire Category theorem which are not equivalent over the base system RCAo. We show that one version (B.C.T.I) is provable in RCAo while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of Z2, which we call RCA' and WKL', and show that RCA' suffices to prove B.C.T...

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi–copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi–copulas. Moreover, it is proved that copulas are nowhere dense in the clas...

We study continuous descent methods for minimizing convex functions defined on general Banach spaces and prove that most of them (in the sense of Baire category) converge.

In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (1) The axiom of dependent choice. (2) Products of compact Hausdorff spaces are Baire. (3) Products of pseudocompact spaces are Baire. (4) Products of countably compact, regular spaces are Baire. (5) Products of regular-closed spaces are Baire. (6) Products of Čech-complete...

Let k be the least cardinal such that the real Une can be covered by k many nowhere dense sets. We show that k can be characterized as the least cardinal such that "infinitely equal" reals fail to exist for families of cardinality k. Let Baire (k) stand for: "The real line is not the union of k many nowhere dense sets (a set is nowhere dense iff its closure has no interior)". The property was e...

A space is a Bake space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely met&able linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E x F is not. If X is a completely met&able linear space of weight K, then X is the direct sum E@F of t...