On Mazurkiewicz's sets and regularity properties of sigma-ideals

The talk will present results from a joint paper with Roman Pol.

Let f:X --> Y be a continuous surjection of the compact metrizable space X onto an uncountable compact metrizable space Y. A Mazurkiewicz set for f is a G_{delta sigma}-set M in X which is a partial selector for f and each G_delta-set in X containing M contains also some fiber of f.

We shall recall the Mazurkiewicz's construction of such a set and apply it to the following effect:

Theorem. Let X be a compact metrizable space and let J be a sigma-ideal of Borel sets in X (containing all singletons) such that every Borel set not in J contains a compact set not in J and the collection of compact sets in J is coanalytic in the hyperspace of compact sets in X.  If J is not ccc, then there is a G_{delta sigma}-set M in X such that:
(i) for no G_delta-set G in X containing M the set G \ M is in J,
(ii) J can be extended to a sigma-ideal  J' of Borel sets in X such that M is in J' but no G_delta-set G in J' contains M.

This can be further specified for sigma-ideals of Borel null-sets of measures and capacities.