Some Links Between Topology, Set Theory and Analysis
by
Dietrich Koelzow
University of Erlangen, Germany

1. A unified approach to continuous and measurable selections of correspondences, due to G. Mägerl (1978)

   Topological and measurable spaces are considered as special paved spaces, and continuous maps as well as semi-continuous correspondences as measurable, with respect to a special paving. Two parameters, \kappa and \alpha, are used to define (for a given paving) certain properties of the domain X, the range Y, and a hull operation on Y. These properties are the basis for an abstract selection theorem, from which (by spezialization of the paving, the hull operation, and the parameters) all selection theorems, in question, follow. The main assumption on X is its (\kappa, \alpha)-paracompactness.

2. Parametrization of projective relations, due to R. Funke (1985)

   Let \Gamma denote an arbitrary projective class, and \Delta(\Gamma) the intersection of \Gamma with its dual class. Under the assumption of determinacy for \Delta(\Gamma), it is shown that every relation in \Gamma [in \Delta(\Gamma)] with fibers of second category possesses an injective parametrization in \Gamma [in \Delta(\Gamma)], having 2^\omega as parameter space. As a guide for this result, the theorem of Y. N. Moschovakis on the uniformization of projective relation is taken. In view of the theorem of D. Martin on the determinacy of the Borel class, the theorem of D. Mauldin on Borel parametrization follows.

3. An analysis of the set of Liouville numbers, due to Chr. Bluhm (2000)

   On the set L of Liouville numbers, the existence of a positive Rajchman measure (i.e. a probability measure, whose Fourier transform tends to zero at infinity) is proven. From this is derived that L is an M-set, but not an H-set. The existence of a Rajchman measure on L is based on the construction of a Cantor set, which is (up to its rationals) contained in L, and which has suitable fourier-dimensional properties. In the construction of the Cantor set, the prime number theorem is used, following an idea of R. Kaufman.

Date received: August 20, 2002