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

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.