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The Axiom of Projective Determinacy implies the existence of a universal set for every n ≥ 1. Assuming there exists a universal set. In ZFC there is a universal set for every α.

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.

If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

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Universal sets for pointsets properly on the nth level of the projective hierarchy

Gregory trees, the continuum, and Martin's axiom

Uncountable Discrete Sets in Extensions and Metrizability

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3 results

Universal sets for pointsets properly on the nth level of the projective hierarchy

Gregory trees, the continuum, and Martin's axiom

Uncountable Discrete Sets in Extensions and Metrizability