Simple closed curves contained in ε-boundaries of planar sets

• Patrakeev, Mikhail ^[1] ; Volkov, Aleksei ^[2]
  1. [1] Ural Branch of the Russian Academy of Sciences

     Ural Branch of the Russian Academy of Sciences

     Rusia

     
  2. [2] Ural Federal University

     Ural Federal University

     Rusia

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 1, 2026
• Idioma: inglés
• DOI: 10.4995/agt.24499
• Enlaces
  • Texto completo
• Resumen

  • The ε-boundary of a set A ⊆ R2 is the set { p ∈ R2 : ρ(p,A) = ε } , where ρ is the Euclidean distance.We prove that if A,B ⊆ R2 are nonempty, connected sets, A is bounded, and 0< ε < ρ(A,B), then the ε-boundary of A contains a simple closed curve (aka a Jordan curve) that separates A and B.This statement follows from the theorem which says that if ε>0 and A ⊆ R2 is a nonempty, bounded, connected set, then the boundary of each component of { p ∈ R2 : ρ(p,A) > ε } is a simple closed curve.Another corollary of this theorem is that the ε-boundary of a nonempty, bounded, connected set A ⊆ R2 contains a simple closed curve bounding the domain that contains the open ε-neighbourhood of A.In all these statements the connectivity condition can be significantly weakened.We also show that, for all ε>0, the ε-boundary of a nonempty, bounded set A ⊆ R2 contains a simple closed curve.

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