For the three years 1929 through 1931 Zermelo was supported by a fellowship provided by the Notgemeinschaft der Deutschen Wissenschaft, the Emergency Association of German Science. This association was founded in 1920 by the leading German academies and scientific organizations in order to avert a collapse of research in the economically disastrous time after the First World War. With a Swiss p...

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture. The normal Moore space conjecture states that every normal Moore space is metrizable. This is known to be not provable from the usual axioms of set theory, since Silver [4] shows that Martin's ax...

A topological space X is said to be completely regular if whenever F is a nonempty closed set and x ∈ X \F , there is a continuous function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1}. A completely regular space need not be Hausdorff. For example, ifX is any set with more than one point, then the trivial topology, in which the only closed sets are ∅ and X, is vacuously completely regular,...

In the present paper, we introduce the fuzzy Nehring axiom, fuzzy Sen axiom and weaker form of the weak fuzzycongruence axiom. We establish interrelations between these axioms and their relation with fuzzy Chernoff axiom. Weexpress full rationality of a fuzzy choice function using these axioms along with the fuzzy Chernoff axiom.

In this paper, given a nonadditive measure ? , topology, which is compatible with convergence in -measure, defined on the real linear space of all (equivalence classes of) measurable functions by using distance introduced Dunford and Schwartz. Some fundamental properties such as open sphere condition, Hausdorff separation axiom, separability, approximation continuous functions, linearlization, ...

A recently proposed axiom system for André’s central translation structures is improved upon. First, one of its axioms turns out to be dependent (derivable from the other axioms). Without this axiom, the axiom system is indeed independent. Second, whereas most of the original independence models were infinite, finite independence models are available. Moreover, the independence proof for one of...

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice. Introduction and Terminology This is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also [vd], [gt], [wgt], and [hkrr]. Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions ...

We consider a class of non-conformal expanding maps on the d-dimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius ε have a mea...

Abstract We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those level $\omega _1$ (that is, ones that are closed under Borel preimages) and iteratively applying operations expansion separated differences. The proof is essentially due to Louveau, it yields at same time a new theorem Van Wesep (namely, expressed as result Hausdorf...

In 1924 Banach and Tarski, using ideas of Hausdorff, proved that there is a partition of the unit sphere S2 into sets A1,...,Ak,B1,..., Bl and a collection of isometries [sigma1,..., sigmak, rho1,..., rhol] so that [sigma1A1,..., sigmakAk] and [rho1B1,..., rholBl] both are partitions of S2. The sets in these partitions are constructed by using the axiom of choice and cannot all be Lebesgue meas...