Rodrigo Hernandez: Wijsman hyperspaces of non-separable metric spaces

10/January/2014,  13:30–15:00

Fields institute, Room 210

Speaker: Rodrigo Hernandez

Talk Title: Wijsman hyperspaces of non-separable metric spaces

Abstract:

The hyperspace CL(X) of a topological space X (at least T1) is the set of all non-empty closed subsets of X. The usual choice for a topology in CL(X) is the Vietoris topology, which has been widely studied. However, in this talk we will consider the Wijsman topology on CL(X), which is defined when (X,d) is a metric space. The Wijsman topology is coarser than the Vietoris topology and in fact it depends on the metric d, not just on the topology. The problem we will address is that of normality of the Wijsman hyperspace. It is known since the 70s that the Vietoris hyperspace is normal if and only if X is compact. But a characterization of normality of the Wijsman hyperspace is still not known. It is conjectured that the Wijsman hyperspace if normal if and only if the space X is separable. Jointly with Paul Szeptycki, we have proved that if X is locally separable and of uncountable weight, then the Wijsman hyperspace is not normal.

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