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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The dimension of closed sets in the Stone-Čech compactification
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by James Keesling PDF
Trans. Amer. Math. Soc. 299 (1987), 413-428 Request permission

Abstract:

In this paper properties of compacta $K$ in $\beta X\backslash X$ are studied for Lindelöf spaces $X$. If ${\operatorname {dim}} K = \infty$, then there is a mapping $f:K \to {T^c}$ such that $f$ is onto and every mapping homotopic to $f$ is onto. This implies that there is an essential family for $K$ consisting of $c$ disjoint pairs of closed sets. It also implies that if $K = \cup \left \{ {{K_\alpha }|\alpha < c} \right \}$ with each ${K_\alpha }$ closed, then there is a $\beta$ such that ${\operatorname {dim}} {K_\beta } = \infty$. Assume $K$ is a compactum in $\beta X\backslash X$ as above. Then if ${\operatorname {dim}} K = n$, there is a closed set $K’$ in $K$ such that ${\operatorname {dim}} K’ = n$ and such that every nonempty ${G_\delta }$-set in $K’$ contains an $n$-dimensional compactum. This holds for $n$ finite or infinite. If ${\operatorname {dim}} K = n$ and $K = \cup \left \{ {{K_\alpha }|\alpha < {\omega _1}} \right \}$ with each ${K_\alpha }$ closed, then there must be a $\beta$ such that ${\operatorname {dim}} {K_\beta } = n$.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 299 (1987), 413-428
  • MSC: Primary 54D35; Secondary 54D40, 54F45
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0869420-3
  • MathSciNet review: 869420