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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Some spaces related to topological inequalities proven by the Erdős-Rado theorem
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by William G. Fleissner PDF
Proc. Amer. Math. Soc. 71 (1978), 313-320 Request permission

Abstract:

The Erdös-Rado theorem is very useful in proving cardinal inequalities in topology. It has been suggested that certain of these inequalities might be strengthened. We note that trees constructed by Jensen and Gregory using various extra axioms of set theory yield several counterexamples to these suggestions; for example, a space $X,|X| = {\omega _2},c(X) = {\omega _1},\chi (X) = \omega$, answering a question of Hajnal and Juhász. We consider the apparently similar relation between $|X|,e(X)$, and $d\Psi (X)$ of Ginsburg and Woods. Using combinatorial consequences of $V = L$, we construct ${G_\delta }$ tree families, and establish that, assuming $V = L$, an infinite cardinal $\kappa$ is weakly compact ${\text {iff}} d\Psi (X) < \kappa ,{e_a}(X) \subset \kappa {\text {imply}}|X| < \kappa$. We consider products of countable chain condition spaces, and show that, using Cohen forcing that (${2^\omega }$ can be anything allowed by König’s theorem and there are spaces $X,Y,c(X) = c(Y) = \omega ,c(X \times Y) = {2^\omega }$). A variation is a space W with the property $c({W^n}) = {\omega _{n - 1}}$.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 71 (1978), 313-320
  • MSC: Primary 54A25; Secondary 02K05
  • DOI: https://doi.org/10.1090/S0002-9939-1978-0493930-4
  • MathSciNet review: 0493930