Metric spaces, Borel and analytic sets, Baire property and measurability, dichotomies, equivalence relations. Possible topics: determinacy, group actions, rigidity, turbulence.

We show that small-bound isomorphisms of spaces affine continuous functions on Choquet simplices with Lindel?f boundaries induce Baire measurable bijections their sets extreme points.

Let X be a Polish space, that is, a separable, completely metrizable topological space. We always assume that X is non-empty and perfect, that is, X has no isolated points. An old theorem of Mycielski (see [8]) says that given a sequence of meager sets …Bn Í X k…n†: n < q†, where 1 < k…n† < q, there exists a perfect set P Í X such that P is free for each Bn , that is, P k…n† Ç Bn Í f…x0 ; . . ....

The concept of a bipolar metric space is presented in this article. In fuzzy space, the notions continuous t-norms and symmetry $$t_s$$ -conorms employs important role. Requirements triangular norms are used to extrapolate with probability density function triangle inequality. Dual processes known as conorms. Continuous define paper. Besides, number topological functional properties have been s...

We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the moduli of continuity.

Proof One follows the argument in [1, pp. 432 – 433] (cf. also [2, Sec. 9]) to show that for each fixed badly approximable (k − 2)-tuple (α1, . . . , αk−2) there is a set of second Baire category of (αk−1, αk) ∈ T2 such that conditions (i), (ii), and (iii) of Theorem 1.7 hold for α = (α1, . . . , αk). Because the set of badly approximable (k − 2)-tuples is dense in Tk−2, and the set of second B...

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...

Given f : X → R ∪ {+∞} a convex and lower semi-continuous function defined on a reflexive Banach space X, and L, a closed linear manifold of X over which f takes at least a real value, the aim of this note is to prove the following Baire category result: in the Euclidean setting, the set of affine functions dominated by f on L for which there is no dominated extension to X is always of first Ba...

We give a constructive proof that Baire space embeds in any inhabited locally non-compact complete separable metric space, X, in such a way that every sequentially continuous function from Baire space to Z extends to a function from X to R. As an application, we show that, in the presence of certain choice and continuity principles, the statement “all functions from X to R is continuous” is fal...