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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1977-0442876-5
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 <I>T</I>, inverse limit, compact Hausdorff space
Article copyright: © Copyright 1977 American Mathematical Society