The following is a list of the definitions used in this paper, more or less in order of appearance.

1. A hyperplane in Eⁿ is a subset H of Eⁿ defined by the set of all ordered n-tuples (x₁, x₂, ..., x_n) such that a₀ + a₁x₁ + a₂x₂ + ... + a_nx_n = 0, for constants a_i, i = 0, 1, 2, ..., n. In E² the hyperplanes are lines and in E³ the hyperplanes are planes.

2. A path connected set A in E² has the two-piece property (TPP) if and only if the intersection of H and A is path connected for any closed half-plane H in E².

3. A 2-manifold-with-boundary M² embedded in E² is a closed point set M² which can be expressed as a union of two sets: M²º, the interior of M² = {p in M² such that there is a disc neighborhood B² of p in E² with B² contained in M²} and the boundary of M² = {q in M² such that there is a neighborhood B² of q in E² with the intersection of B² with M² given by the 1-1 continuous image of the intersection of an open disc about the origin with the closed upper half-plane, with q corresponding to the origin}.

4. A set A has the circular two-piece property (CTPP) if the intersection of A with closure(D_i) is path connected, for either complementary component D_i of any circle or straight line S in E².

5. A grid square is a subset S of R² with center (x₀, y₀) and radius r defined by

S = {(x, y): max(|x - x₀|, |y - y₀|) = r}.

We denote the interior of S by D₁ and the exterior by D₂.

6. A subset A of R² has the Grid-Square Two-Piece Property (GTPP) if for every grid square S the intersection of A with closure(D_i) is path connected, i = 1, 2.

7. An n-dimensional grid cube is a subset S of Rⁿ with center c = (c₀, c₁, ..., c_n) and radius r defined by

S = {x = (x₀, x₁, ..., x_n): max_i |x_i - c_i| = r}.

We again denote the interior of S by D₁ and the exterior by D₂.

8. A subset A of Rⁿ has the Grid-Cube Two-Piece Property (GTPP) if for every grid cube S the intersection of A with closure(D_i) is path connected, i = 1, 2.