On Unions and Intersections of Simply Connected Planar Sets

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–Grunbaum–Klee, related to Molnar’s result on intersections of simply connected planar sets.