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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Strong homology and the proper forcing axiom


Authors: Alan Dow, Petr Simon and Jerry E. Vaughan
Journal: Proc. Amer. Math. Soc. 106 (1989), 821-828
MSC: Primary 55P55; Secondary 03E35, 55N07
MathSciNet review: 961403
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Abstract: This paper concerns applications of set theory to the problem of calculating the strong homology of certain subsets of Euclidean spaces. We prove the set theoretic result that it is consistent that every almost coinciding family indexed by $ ^\omega \omega $ is trivial (e.g., the proper forcing axiom implies this). This result, combined with results of S. Mardešić and A. Prasalov, show that the statement "the $ k$-dimensional strong homology of $ {Y^{(k + 1)}}$ (the discrete sum of countably many copies of the $ (k + 1)$-dimensional Hawaiian earring) is trivial" is consistent with and independent of the usual axioms of set theory.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0961403-5
Keywords: Strong homology, Hawaiian earring, almost coinciding families, continuum hypothesis, bounding number, dominating number, $ (\kappa ,\kappa )$-gaps in the sense of Hausdorff, $ {\aleph _1}$-scale
Article copyright: © Copyright 1989 American Mathematical Society