Characterizing global properties in inverse limits
The inverse limit X of an inverse sequence (inverse system indexed by the positive integers) \(\{X_ i,P_{i+1,i}\}\) of compact metric spaces, may fail to be locally connected (n-LC for some or all n) even when the bonding maps are sufficiently nice and each \(X_ i\) is a manifold. For instance, consider the 2-adic solenoid as the inverse limit of an inverse sequence where \(X_ i\) is the circle and the bonding map is the 2-fold covering map. The author gives several necessary and sufficient conditions on X which suitably insure that X is, for instance, an absolute neighborhood retract, \(LC^ n\) (locally connected) up to dimension n, or n-dimensional, the results and related techniques are motivated by shape theory. The paper contains many other results which are too technical to be included in this brief review.
scientific article; zbMATH DE number 3870212