An Understanding The Application Of Helly Theorem For A Topology Of R²

Let to be simple polygons in the plane. If all double and triple intersections of X_i are also simple polygons and non empty, then the intersection of all X_i is also a simple polygon [3] (Figure 1).

Definition

A simple polygon is defined to be connected without holes, simply connected finite union of convex polygons.

The definition and description are based on “A Helly-type Theorem for simple polygons” and “A Helly-type theorem for Intersections of compact connected sets in the plane” by Marilyn Breen [1,2].

Definition

A planar set S is simply connected if for every simple closed curve λ in S and every point q in the bounded region determined by the λ, q belongs to S (Figure 2).

While the convex set allows only straight lines, the path connected in the simple polygon can be any curves for two distinct points. As simple polygons are not necessarily convex sets, the classic Helly theorem is not applicable. However, we proceed to use compact sets and theorems to show for the intersection of simple polygons.

Theorem: (Molnar’s theorem) If C = {C₁...,C_n} is a family of simply connected compact sets in the plane such that every two (not necessarily distinct) sets of C have a nonempty intersection, then and simply connected.

Theorem: (98’) ⇔ (the main theorem) • Every two (not necessarily distinct) sets of C have a connected union and every three (not necessarily distinct) sets of C have a simply connected union.
• C is a family of simply connected sets such that every two (not necessarily distinct) sets of C have a connected intersection and every three (not necessarily distinct) sets of C have a nonempty intersection.

Part (1) implies (2) using Lemma 1 and Lemma 2. Part (2) implies (1) using Proposition 1 applied by the Lemma 3.

a. Lemma 1: Let C₁ and C₂ to be compact connected sets in the plane. If C₁ ∪C₂ is simply connected.
b. Lemma 2: For C₁, C₂, C₃ nonempty compact connected sets in the plane, if every two of these sets have a connected union and all three sets have a simply connected union, then C₁ ∩C₂ ∩C₃ ≠ ∅.
c. Lemma 3: Let λ be a simple closed curve in the plane, with point q in the bounded region determined by λ and point r in the unbounded region determined by λ.

Let μ₁, μ₂, μ₃ be simple closed curves from q to r, with λ ∩μ_i finite for 1≤ i ≤ 3 and λ ∩μ_i ∩μ_j ≠ ∅ , for 1≤ i ≤ j ≤ 3 . Then, for some point b₁ ∈μ₁ ∩λ is in a bounded region determined by μ₂ ∪μ₃

In summary, for three different closed curve connecting points q∈λ to r we are able to find a point b₁ that is the intersection of μ₁ and λ bounded inside the connected region of λ_i ∪λ_j . Note that the point b_i from λ ∩μ_i does not belong to λ ∩μ_i ∩μ_j ≠ ∅ . This means there are three cases to find the point b_i from μ_i ∩λ defined in μ₂ ∪μ₃ or μ₂ ∩λ defined in μ₁ ∪μ₃ or μ₃ ∩λ defined in μ₁ ∪μ₂

d. Proposition 1: For family of set C of n ≥ 2 in the plane, if every two sets of C have a connected, simply connected union and is simply connected.

The goal of Proposition 1 with Lemma 3 is to show for three sets with Part (2) to be simply connected union of Part (1).

Theorem: (An intersection theorem)

Let P be the family of simply polygons P₁,…, P_n in the plane. If every three (not necessarily distinct) polygons of P have a simply connected union and every two polygons of P have a nonempty intersection, the .

The intersection theorem could not directly apply to simple polygons to derive a common intersection as every three polygons are simply connected union for a given condition.

Difference

While the Molnar’s theorem applicable for compact connected sets and an intersection theorem is applicable for simple polygons. Generally, simple polygons in the plane are identified to be compact connected sets.

While the Molnar’s theorem is conditioned for triple intersection of sets to be nonempty, the intersection theorem is conditioned for triple sets to be simply connected union.

In the Molnar’s theorem, is simply connected, the intersection theorem does not support to be simply connected.

Similarity

Using the induction, both theorems resulted in the common intersection of finitely many sets to be non-empty. Note that the intersection theorem is a strong version of theorem 1 with the same conclusion for Molnar’s theorem of common intersection for finitely many sets.

With loss of generality, the intersection theorem where all triples polygons have simply connected union is stronger than Part (2) with a condition that all triple sets of intersection is nonempty.

Corollary

Based on the Molnar’s theorem and theorem 1 of part (1), is nonempty, connected and simply connected and is simply connected.

Basis

The conclusion that , is supported by the Molnar’s theorem as is nonempty, simply connected, which is based on the induction. However, is connected, simply connected is supported by the theorem 1 of part (1) for induction with proposition 1 to be applied with.

Consequence

This corollary proves that , is a finite union of convex sets for simple polygons of X1,…,Xn. This fact that is a simple polygon is supported by the corollary only for the basis of main theorem and Molnar’s theorem where a simple polygon consists of finite union of convex polygons.

None.

No conflict of interest.