Monotoneity Relative to a Point and Inverse Limits of Continua

It is shown that the inverse limit of an inverse system of smooth continua is a smooth continuum provided that there exists a thread composed of initial points and that the bonding mappings are monotone relative to these points. It also is proved that the property of being an arboroid, a dendroid, a generalized tree, a fan or a smooth fan is an invariant under the inverse limit operation if the above assumptions are satisfied for the inverse systems. The aim of this paper is to investigate some properties of mappings which are monotone relative to a point [18]. After preliminaries, in § 2 we prove some necessary and sufficient conditions under which a mapping f : X -+ Y is monotone relative to a point P EX. The conditions are formulated in terms of some quasi-orders on X and on Y, and are applied in the further results. The rest of the paper is devoted to the inverse limits. In § 3 it is shown that smoothness of continua (in the sense of Gordh [10]) is preserved under the inverse limit operation if the bonding mappings are monotone relative to points which form a thread, and that the property of being an arboroid (a dendroid, a generalized tree, a fan, a smooth fan) is also an invariant of this operation, while the property of being a dendrite is not (§ 4). The fourth paragraph contains some examples showing that the assumptions concerning the inverse systems are essential in the theorems proved in § 3. Further, some problems, in particular related to contractibility of continua and to the existence of a continuous selection on the family C (X) of subcontinua of a given continuum X are asked. The authors thank the referee for pointing out some inaccuracies in an earlier version of this article.