Simple closed curves contained in ε-boundaries of planar sets

Key Points

• The research focuses on proving the existence of simple closed curves in the epsilon-boundaries of bounded, connected sets in the plane.
• Proved the existence of a simple closed curve for ε-boundaries given nonempty, bounded, connected sets A and B in R2.
• Demonstrated that the boundary of each component of points with distance greater than ε from A is a simple closed curve.
• Examined the weakening of connectivity conditions for these results.
• For 0<ε<ρ(A,B), the ε-boundary of A contains a simple closed curve that separates A and B.
• The ε-boundary always contains a simple closed curve bounding the open ε-neighborhood of A for all ε>0.
• The connectivity condition for these theorems can be significantly weakened.