Inverse limits and smoothness of continua

Abstract

Relations between the inverse limit operation and smoothness of continua are studied in this paper. Inverse systems with monotone bonding mappings are mainly discussed. It is shown that smoothness of continua is preserved under the inverse limits of inverse systems with monotone bonding mappings provided there exists a thread composed of points at which the factor spaces are smooth; in parti­ cular it follows that the inverse limit of an inverse sequence of smooth dendroids with monotone bonding mappings is a smooth dendroid if the corresponding thread does exist. Topological spaces considered throughout this paper are assumed to be compact (thus Hausdorff. see [7]. p. 165) and the mappings are assumed to be continuous. By a continuum we mean a compact connected space. The following notation will be used. {X)..f).l', A} denotes an inverse system of the topological spaces X). with continuous bonding mappings f).l': XI'-X). for any A~J1., where A, J1.EA and A is a set directed by the relation ~. We assume that fU is the identity, and we denote by X=lim {X).,f).l', A} the inverse limit space. Further, 1t).: X-X). denotes the projection from the inverse limit space into the A-th factor space. In a particular case when A is the set N of natural numbers with the natural ordering ~ we write {Xi,F}i:l and 1t1 instead of {X).,fAi', A} and 1t). respectively, where F: Xi+l_Xi are bonding maps, and then {Xi,!')i:l is caned an inverse sequence. Given a point pEX = lim {X).,P", A}, we put P).=1t).(p)EX). and we write p={p).}. If A=N, we write p={pl}. Obviously we have