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 α , f α β , α , β ∈ I } \{ {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 ∈ X 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 α ∈ I \alpha \in I and disjoint sets U and V open in X α {X_\alpha } , neighborhoods respectively of p α {p_\alpha } and A α {A_\alpha } such that for all β ⩾ α , f α β − 1 ( U ) \beta \geqslant \alpha ,f_{\alpha \beta }^{ - 1}(U) lies in a single component of X β − f α β − 1 ( V ) {X_\beta } - f_{\alpha \beta }^{ - 1}(V) . This is Theorem B of the text.