On Unions and Intersections of Simply Connected Planar Sets

Abstract

Abstract. We construct several simple examples of planar compacta which show that without additional conditions, a theorem of Breen and a direct generalization of the Seifert-van Kampen theorem fail. We give answers to two conjectures of Bogatyi and a partial solution to his third conjecture. We give a counterexample to a statement in the classical survey paper by Danzer-Gr€ u unbaum-Klee, related to Molnár's result on intersections of simply connected planar sets.2000 Mathematics Subject Classification: 54F15, 55N10, 54D05

A Krasnosel’skii theorem for orthogonal polygons starshaped via staircase (n + 1)-paths

“…Proof. The result follows from the version of Molnár's theorem by Karimov et al [10], comments in [1], and Theorem 1.…”

“…We will show that 1. every two of the V sets have a path connected intersection, and 2. every three of the V sets have a nonempty intersection. Then using a version of Molnár's theorem by Karimov, Repovs, andZeljko [10], ∩{V t : t in S} = ∅. For any point z in this intersection, z Ker n+1 S. Part 1.…”

“…Since every two of the V sets have a path connected intersection and every three have a nonempty intersection, we may apply a version of Molnár's theorem by Karimov et al [10] to conclude that ∩{V t : t in S} = φ. For any point z in this intersection, z sees each point t of S via staircase (n + 1)-paths.…”

Sweep data of electrical impedance tomography

“…This is a contradiction, which shows that is composed of a finite number of path connected components. To complete the proof, we note that every path connected component of = ∩ is simply connected since the fundamental group of the intersection of any two simply connected planar sets is trivial (see, e.g., [18]). Now we have gathered enough tools to introduce the convex sweep support and show that it carries some useful properties.…”