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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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The $\textrm {SUP=MAX}$ problem for $\delta$
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by Andrew J. Berner and István Juhász PDF
Proc. Amer. Math. Soc. 99 (1987), 585-588 Request permission

Abstract:

Let $\delta \left ( X \right ) = \operatorname {sup}\{ d(D):D$ is a dense subspace of $X\}$. It is shown that if $\kappa$ is a limit cardinal, but not a strong limit, and ${\text {cf}}\left ( \kappa \right ) > \omega$, then there is a $0$-dimensional Hausdorff space $X$ such that $\delta \left ( X \right ) = \kappa$, but for all dense $D \subset X,d(D) < \kappa$. For all other values of $\kappa$, if $X$ is Hausdorff and $\delta \left ( X \right ) = \kappa$, then there is a dense $D \subset X$ such that $d\left ( D \right ) = \kappa$.
References
    István Juhász, Cardinal functions—Ten years later, Math. Centre Tract 123, Amsterdam, 1980.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 99 (1987), 585-588
  • MSC: Primary 54A25
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0875405-9
  • MathSciNet review: 875405