Borel sets in compact spaces: some Hurewicz type theorems

HOMOTOPICAL EXCISION, AND HUREWICZ THEOREMS, FOR n-CUBES OF SPACES

where e is induced by the ‘excision’ inclusion. That e is an isomorphism in a range of dimensions is shown by the classical Blakers-Massey triad connectivity theorem: if A, B, and A∩B are connected, {A,B} is an open cover of X and (A, A ∩ B) is p-connected, (B, A ∩ B) is q-connected, then the triad (X; A,B) is (p + q)-connected. (See, for example, [11, p. 211].) Further, if p, q > 2 and π1(A∩B)...

Borel Theorems for Random Matrices from the Classical Compact Symmetric Spaces

Abstract. We study random vectors of the form (Tr(AV ), . . . ,Tr(AV )), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et ...

Absolute Borel Sets and Function Spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Froĺık for characterizing Čechcomplete spaces. We also show that the absolute Borel class of X is determined by the uniform structure of the space of continuous functions Cp(X); however the case of absolut...

On Discrete Borel Spaces and Projective Sets

Let J denote the unit interval, S—IXI the unit square; Cj and Cs the class of all subsets of I and 6, respectively. By Cj X Cj is meant the or-algebra on S generated by rectangles with sides in C/. The purpose of this note is to prove the following theorem (which settles a problem of S. M. Ulam) and observe some of its consequences. Without explicit mention, the axiom of choice has been assumed...

Common Fixed Point Theorems of Integral Type in Modular Spaces