We demonstrate a relationships between the representation theory of Borel subgroups and parabolic subgroups of general linear groups. In particular, we show that the representations of Borel subgroups could be computed from representations of certain maximal parabolic subgroups.

We show that the set of absolutely normal numbers is Π3-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π3-complete in the effective Borel hierarchy.

We study א0-stable theories, and prove that if T either has eniDOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of λ-Borel completeness and prove that such theories are λ-Borel complete. Using this, we conclude that an א0-stable theory satisfies I∞,א0(T, λ) = 2 λ for all cardinals λ if and only if T either has eni-DOP or is eni-deep.

We consider Borel sets of the form A ! (with usual topology) where cardinality of is less than some uncountable regular cardinal. We obtain a \normal form" of A, by nding a Borel set () such that A and () continuously reduce to each other. We do so by deening Borel operations which are homomorphic to the rst Veblen ordinal functions of base required to compute the Wadge degree of the set A: the...

We investigate the connection between the Borel class of a function f and the Borel complexity of the set C(f) = {C ∈ J(X) : f |C is continuous} where J(X) denotes the compact subsets of X with the Hausdorff metric. For example, we show that for a function f : X → Y between Polish spaces; if C(f) is Fσδ in J(X), then f is Borel class one.