Projective σ-compactness, ω1-caliber, and Cp-spaces

Determinacy from Strong Compactness of Ω1

In the absence of the Axiom of Choice, the “small” cardinal ω1 can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ω1 is Xstrongly compact (where X is any set) if there is a fine, countably complete measure on ℘ω1(X). Working in ZF + DC, we prove that the ℘(ω1)-strong compact...

On effective σ-boundedness and σ-compactness

We prove several dichotomy theorems which extend some known results on σ -bounded and σ -compactpointsets. In particular we show that, given a finite number of Δ1 equivalence relations F1 , . . . , Fn , any Σ1 set A of the Baire space either is covered by compact Δ1 sets and lightface Δ1 equivalence classes of the relations Fi , or A contains a superperfect subset which is pairwise Fi -inequiva...

Locally Compact, Ω1-compact Spaces

This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo large cardinals) that a locally compact space X must be the union of countably many ω-bounded subspaces if every closed discrete subspace of X is countable [in other words, if X is ω1-compact]? A space is ω-bounded if every countable subset has compact closure. This is a strengthening of countable ...

Biquaternions Lie Algebra and Complex-Projective Spaces

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σ-Normality of Topological Spaces