Conjuntos de Cantor, Números de Liouville y Números Diofantinos

Abstract

In this document we study the existence of two Cantor sets, one contained in the Liouville Numbers (L) and the other one in the Diophantine Numbers (D). First, by using some properties of L, we built an uncountable and closed set S⊆ L. By Bendixson’s Theorem, there exists a perfect set C1⊆ S⊆ L. Since L is a null set (λ(L)=0), C1 is also a null set. We conclude that C1⊆L is a Cantor set. On the other hand, from the fact that D is a countable union of closed and nowhere dense sets, and since D is uncountable, there exists D0⊆ D such that D0 is an uncountable closed and nowhere dense set. By Bendixson’s Theorem, there is a perfect set C2⊆ D0⊆D. Since D0 is a nowhere dense set, we conclude C2⊆ D is a Cantor set.