Geometric taming of compacta in 𝐸ⁿ

Abstract

We investigate k k -dimensional compacta in E n ( k ⩽ n − 3 ) {E^n}(k \leqslant n - 3) that satisfy geometric properties. We prove that such a compactum X X in E n {E^n} is tamely embedded if each point of X X can be touched by the tip of a cone from the complement of X X . Furthermore, we show that a k k -dimensional compactum Y Y in E n ( k ⩽ n − 3 ) {E^n}(k \leqslant n - 3) is tame if Y Y has vertical order n − k − 2 n - k - 2 .