Topological n-cells and Hilbert cubes in inverse limits

Abstract

It has been shown by S. Mardešić that if a compact metrizable space X has dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then X must contain an arc. We are going to prove that if X = (|Ka|,pba,(A,))is an inverse system in set theory of triangulated polyhedra|Ka|with simplicial bonding functions pba and X = lim X, then there exists a uniquely determined sub-inverse system XX= (|La|, pba|Lb|,(A,)) of X where for each a, La is a subcomplex of Ka, each pba|Lb|:|Lb| → |La| is surjective, and lim XX = X. We shall use this to generalize the Mardešić result by characterizing when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell and do the same in the case of an inverse system of finite triangulated polyhedra with simplicial bonding maps. We shall also characterize when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain an embedded copy of the Hilbert cube. In each of the above settings, all the polyhedra have the weak topology or all have the metric topology(these topologies being identical when the polyhedra are finite).

Keywords

• Hilbert cube
• inverse limit
• inverse sequence
• inverse system
• polyhedron
• simplicial inverse system
• simplicial map
• topological n-cell
• triangulation