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On the dimension of limits of inverse systems


Author: Yukinobu Yajima
Journal: Proc. Amer. Math. Soc. 91 (1984), 461-466
MSC: Primary 54F45; Secondary 54B10
DOI: https://doi.org/10.1090/S0002-9939-1984-0744649-6
MathSciNet review: 744649
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Abstract: We say that the limit of an inverse system $ X = \underleftarrow {\lim }\left\{ {{X_\lambda },\pi _\mu ^\lambda ,\Lambda } \right\}$ is cylindrical if each finite cozero cover of $ X$ has a $ \sigma $-locally finite refinement consisting of sets of the form $ \pi _\lambda ^{ - 1}(U)$, where $ U$ is a cozero-set in $ {X_\lambda }$ and $ {\pi _\lambda }:X \to {X_\lambda }$ is the projection.

We prove that if $ X$ is cylindrical, then $ \dim X = \sup \left\{ {\dim {X_\lambda }:\lambda \in \Lambda } \right\}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0744649-6
Keywords: Covering dimension, limit of inverse system, cylindrical, cozero cylinder, perforable inverse sequence, Cartesian product, finite subproduct
Article copyright: © Copyright 1984 American Mathematical Society

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