The circular two-piece property (CTPP) is examined in the plane in the first section of [1]. These results are repeated here without proof.

Definition. 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.

We first notice that a circle, a closed disc, and a closed half-plane all have the CTPP. Like with the TPP, we can remove any number of discs (not just any convex sets will work now) and still have the CTPP:

Proposition. If A is a closed disc, a closed half-plane, or the whole space, and if {B^j} is a collection of disjoint open discs in A, then A \ {union B^j} has the CTPP.

In order to show the converse of this proposition, which is the main theorem of the CTPP in the plane, [1] first shows the following three propositions.

Proposition. If A is a simple closed curve in E² with the CTPP, then A is a circle.

Proposition. If A is a region bounded by a simple (closed or infinite) curve and has the CTPP, then the boundary of A is a circle or a straight line.

Proposition. If A is a bounded closed connected CTPP set with unbounded complementary component C, then E² \ C is a closed disc.

Finally, we classify the closed sets in the plane with the CTPP:

Theorem. The only connected closed sets in E² with the CTPP are a closed disc, a closed half-plane, or the whole plane, each with a collection of disjoint open discs removed.

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