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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The dimension of closed sets in the Stone-Čech compactification


Author: James Keesling
Journal: Trans. Amer. Math. Soc. 299 (1987), 413-428
MSC: Primary 54D35; Secondary 54D40, 54F45
MathSciNet review: 869420
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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 }\vert\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 }\vert\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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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869420-3
PII: S 0002-9947(1987)0869420-3
Keywords: Stone-Čech compactification, Lindelöf space, dimension, homotopically onto, $ c$-torus, essential family, sum theorem
Article copyright: © Copyright 1987 American Mathematical Society