On intersections of compacta in Euclidean space

Abstract

Let X {\mathbf {X}} be a codimension-three tame compactum in Euclidean space E n {{\mathbf {E}}^n} . If dim ⁡ X × Y > n \operatorname {dim} {\mathbf {X}} \times {\mathbf {Y}} > n , then every map f : Y → E n f:{\mathbf {Y}} \to {{\mathbf {E}}^n} can be approximated by map g g with X ∩ Im ⁡ g = ∅ {\mathbf {X}} \cap \operatorname {Im} g = \emptyset .