Topological applications of generic huge embeddings

Tall, Franklin D.

doi:10.1090/S0002-9947-1994-1223302-X

Topological applications of generic huge embeddings
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Trans. Amer. Math. Soc. 341 (1994), 45-68 Request permission

Abstract:

In the Foreman-Laver model obtained by huge cardinal collapse, for many $\Phi ,\Phi ({\aleph _1})$ implies $\Phi ({\aleph _2})$. There are a variety of set-theoretic and topological applications, in particular to paracompactness. The key tools are generic huge embeddings and preservation via $\kappa$-centred forcing. We also formulate "potent axioms" à la Foreman which enable us to transfer from ${\aleph _1}$ to all cardinals. One such axiom implies that all ${\aleph _1}$-collectionwise normal Moore spaces are metrizable. It also implies (as does Martin’s Maximum) that a first countable generalized ordered space is hereditarily paracompact iff every subspace of size ${\aleph _1}$ is paracompact.

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