Metrizability of certain countable unions

Descriptive complexity of countable unions of Borel rectangles

We give, for each countable ordinal ξ ≥ 1, an example of a ∆2 countable union of Borel rectangles that cannot be decomposed into countably many Πξ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ∆ξ-measurable countable coloring. 2010 Mathematics Subject Classification. Primary: 03E15, Secon...

Countable Unions of Simple Sets in the Core Model

We follow 8] in asking when a set of ordinals X is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical 1 Skolem function for K can be so decomposed, provided K is such that no !-closed lters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erd} os type property.

On edge-magic labelings of certain disjoint unions of graphs

Metrizability of Decomposition Spaces

Simultaneous Metrizability of Coarse Spaces

A metric space can be naturally endowed with both a topology and a coarse structure. We examine the converse to this. Given a topology and a coarse structure we give necessary and sufficient conditions for the existence of a metric giving rise to both of these. We conclude with an application to the construction of the coarse assembly map.