Large small sets
Le problème des cardinaux singuliers
Level by Level Inequivalence, Strong Compactness, and GCH
We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.
Linear orders and MA + ¬wKH
I prove that the statement that “every linear order of size can be embedded in ” is consistent with MA + ¬ wKH.
Locally compact perfectly normal spaces may all be paracompact
We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...
Locally nice spaces under Martin's axiom
Logique, catégories et faisceaux
Lusin sequences under CH and under Martin's Axiom
Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.