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Proceedings of the American Mathematical Society

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Relationships between continuum neighborhoods in inverse limit spaces and separations in inverse limit sequences

Author: Harvey S. Davis
Journal: Proc. Amer. Math. Soc. 64 (1977), 149-153
MSC: Primary 54B25; Secondary 54F20
MathSciNet review: 0442876
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Abstract: The main result of this paper is the following theorem. Let $ \{ {X_\alpha },{f_{\alpha \beta }},\alpha ,\beta \in I\} $ be an inverse system of compact Hausdorff spaces and continuous onto maps with inverse limit X. Let $ p \in X$ and A be closed in X. There exists a continuum neighborhood of p disjoint from A if and only if there exists $ \alpha \in I$ and disjoint sets U and V open in $ {X_\alpha }$, neighborhoods respectively of $ {p_\alpha }$ and $ {A_\alpha }$ such that for all $ \beta \geqslant \alpha ,f_{\alpha \beta }^{ - 1}(U)$ lies in a single component of $ {X_\beta } - f_{\alpha \beta }^{ - 1}(V)$. This is Theorem B of the text.

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Keywords: Continuum neighborhood, set function T, inverse limit, compact Hausdorff space
Article copyright: © Copyright 1977 American Mathematical Society

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