Petr Holický

Lindelöf modifications and K -analytic spaces

We present simple constructions of spaces which are countably K -determined, Cech analytic and not K -analytic. We prove that the statement “every uncountable K -analytic space contains an uncountable compact subset” is equivalent to b > ω, extending a result of the first author.

ON ENTROPIC REPULSION IN LOW TEMPERATURE ISING MODELS

We investigate the phenomenon of “entropic repulsion” of interfaces in the low temperature Ising model. We study the Gibbs states in the half-space of Z d , d ≥ 3, analogously to the simplified situations of the solid-on-solid model (Bricmont, Mellouki and Frohlich 1986) and the “interface model” (Melichercik 1991). Our method is based on the Pirogov-Sinai theory and on a detailed study of cluster expansions. The crucial step consists in an estimate of some “metastable” partition sums corresponding to different heights of the interface. We formulate the basic estimate (2) and indicate its proof here.

Descriptive classes of sets in nonseparable spaces

Several attempts to transfer methods of the classical descriptive set theory to study measurability in nonseparable metrizable or more general topological spaces are presented. In particular, the theories of K-analytic spaces and of absolute Souslin metrizable spaces give quite interesting results. The spaces which are results of the Souslin operation applied to the families of Borel sets (Čech-analytic spaces) or of resolvable sets in every Tychonoff space allow further generalizations. Many nonseparable Banach spaces with their weak topology may serve as examples of such spaces. We recall a number of results and we point out open questions which seem to make the extension of the classical descriptive set theory and its applications to functional analysis not yet sufficiently complete.ResumenSe presentan varios enfoques para transferir métodos de la teoría clásica descriptiva de conjuntos al estudio de medibilidad en espacios metrizables no separables e incluso en espacios más generales. En particular, las teorías de espacios K-analíticos y de espacios absolutamente Souslin (en espacios metrizables) dan resultados muy interesantes. Otras generalizaciones las proporcionan los espacios obtenidos al aplicar la operación de Souslin a familias de conjuntos de Borel (espacios Čech-analíticos), así como los conjuntos resolubles en cada espacio de Tychonoff. Muchos espacios de Banach no separables provistos con la topología débil suministran ejemplos de estos espacios. Se dan bastantes resultados y se señalan cuestiones abiertas que muestran que la extensión al análisis funcional y a sus aplicaciones de la teoría clásica descriptiva de conjuntos no está todavía suficientemente completa.