On dimension and shape of inverse limits with set-valued functions

Hisao Kato¹

¹ Institute of Mathematics University of Tsukuba Ibaraki 305-8571, Japan

Fundamenta Mathematicae, Tome 236 (2017) no. 1, pp. 83-99.

Résumé

Recently, several topological properties of inverse limits of compacta with upper semicontinuous set-valued functions have been studied by many authors. The study of such inverse limits has developed into one of rich topics of geometric topology. There are many differences between the theory of inverse limits with mappings and the theory with set-valued functions. In this paper, we investigate the dimension and the shape of inverse limits with set-valued functions. To evaluate the dimension of the inverse limit $\varprojlim \{X_i,f_{i,i+1}\}$ of a given inverse sequence $\{X_i,f_{i,i+1}\}_{i=1}^{\infty}$ of compacta with set-valued functions satisfying $$ \dim \{x\in X_{i+1}\mid \dim f_{i,i+1}(x) \geq 1\} \leq 0\quad\ (i\in \mathbb N), $$ we define expand-contract sequences in $\{X_i,f_{i,i+1}\}_{i=1}^{\infty}$ and an index $\tilde{J}(\{X_i,f_{i,i+1}\})$. By use of the index, we prove that $$ \dim \varprojlim \{X_i,f_{i,i+1}\} \leq \tilde{J}(\{X_i,f_{i,i+1}\})+ \sup\{\dim X_i\mid i\in \mathbb N\}. $$ Moreover, we evaluate lower bounds of dimensions of some inverse limits of 1-dimensional compacta with set-valued functions. We study the shape of inverse limits with cell-like set-valued functions.

Détail

DOI : 10.4064/fm233-4-2016

Keywords: recently several topological properties inverse limits compacta upper semicontinuous set valued functions have studied many authors study inverse limits has developed rich topics geometric topology there many differences between theory inverse limits mappings theory set valued functions paper investigate dimension shape inverse limits set valued functions evaluate dimension inverse limit varprojlim given inverse sequence infty compacta set valued functions satisfying dim mid dim geq leq quad mathbb define expand contract sequences infty index tilde index prove dim varprojlim leq tilde sup dim mid mathbb moreover evaluate lower bounds dimensions inverse limits dimensional compacta set valued functions study shape inverse limits cell like set valued functions