Simple closed curves contained in ε-boundaries of planar sets
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Patrakeev, Mikhail
[1]
;
Volkov, Aleksei
[2]
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[1] Ural Branch of the Russian Academy of Sciences
Ural Branch of the Russian Academy of Sciences
Rusia
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[2] Ural Federal University
Ural Federal University
Rusia
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- Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 1, 2026
- Idioma: inglés
- Enlaces
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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.
