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Topological applications of generic huge embeddings


Author: Franklin D. Tall
Journal: Trans. Amer. Math. Soc. 341 (1994), 45-68
MSC: Primary 03E35; Secondary 03E55, 03E75, 54A35
DOI: https://doi.org/10.1090/S0002-9947-1994-1223302-X
MathSciNet review: 1223302
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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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DOI: https://doi.org/10.1090/S0002-9947-1994-1223302-X
Article copyright: © Copyright 1994 American Mathematical Society

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