Unions of cells with applications to visibility

Abstract

A crumpled n-cell C in E n {E^n} is proven to be an n-cell ( n ≠ 4 ) (n \ne 4) when it is known to contain two n-cells C 1 {C_1} and C 2 {C_2} , one of which is flat, such that Bd C ⊂ ( Bd C 1 ) ∪ (Bd C 2 ) {\text {Bd}}\;C \subset ({\text {Bd}}\;{C_1}) \cup {\text {(Bd}}\;{C_2}) . This theorem is applied to show that C is an n-cell if its boundary is the union of two closed sets each of which is seen from some point of Int ⁡ C \operatorname {Int} C . Examples are given to show that flatness of one of C 1 {C_1} and C 2 {C_2} is necessary in the first theorem and to show that two is the largest integer for which either theorem is true.