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Some Existence Results on Cantor Sets Journal of the Egyptian Mathematical Society 25 (2017) 326–330 Contents lists available at ScienceDirect Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems Original Article Some existence results on Cantor sets ∗ Borys Álvarez-Samaniego a, , Wilson P. Álvarez-Samaniego a, Jonathan Ortiz-Castro b a Núcleo de Investigadores Científicos, Facultad de Ingeniería, Ciencias Físicas y Matemática, Universidad Central del Ecuador (UCE), Quito, Ecuador b Facultad de Ciencias, Escuela Politécnica Nacional (EPN), Quito, Ecuador a r t i c l e i n f o a b s t r a c t Article history: The existence of two different Cantor sets, one of them contained in the set of Liouville numbers and the Received 2 November 2016 other one inside the set of Diophantine numbers, is proved. Finally, a necessary and sufficient condition Revised 25 January 2017 for the existence of a Cantor set contained in a subset of the real line is given. Accepted 11 February 2017 Available online 18 March 2017 ©2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. MSC: ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) 54A05 54B05 54A25 Keywords: Cantor set Liouville numbers Diophantine numbers 1. Introduction that q > 1 and p 1 First, we will introduce some basic topological concepts. 0 < α − < . (1.1) q q n Definition 1.1. A nowhere dense set X in a topological space is a If β ∈ R is not a Liouville number, we say that β is a Diophan- set whose closure has empty interior, i.e. int ( X ) = ∅ . tine number. The sets of Liouville and Diophantine numbers are, Definition 1.2. A nonempty set C ⊂ R is a Cantor set if C respectively, denoted by L and D . is nowhere dense and perfect (i.e. C = C , where C := { p ∈ Joseph Liouville, by giving two different proofs, showed the ex- R ; p is an accumulation point of C} is the derived set of C ). istence of transcendental numbers for the first time in 1844 [1,2] . Definition 1.3. A condensation point t of a subset A of a topolog- Later, in 1851 [3] more detailed versions of these proofs were ical space, is any point t , such that every open neighborhood of t given. It was also shown in [3] that the real number contains uncountably many points of A . + ∞ 1 , We denote by Q the set of rational numbers. The symbol Z is 10 k ! k =1 used to denote the set of integers, Z + = { 1 , 2 , . .} denotes the pos- itive integers and N = { 0 , 1 , 2 , . .} is the set of all natural numbers. which was already mentioned in [1] , is a transcendental number. The cardinality of a set B is denoted by | B | . We denote by OR , the It is worth noting that the techniques used in [1–3] allow proving class of all ordinal numbers. Moreover, represents the set of all that all Liouville numbers are transcendental. countable ordinal numbers. Some general properties of the set of Liouville numbers are: L is a null set under the Lebesgue measure (i.e. λ(L ) = 0 ), it is α ∈ R \ Q α Definition 1.4. Let . We say that is a Liouville number a dense G δ set in the real line, L is an uncountable set and, ∈ N , = = , if for all n there exist integer numbers p pn and q qn such more specifically, it has the cardinality of the continuum. Since the Lebesgue measure of the Diophantine numbers is infinity, it is an uncountable set. By using Theorem 3.1 below, there exists a Cantor ∗ Corresponding author. set in D , and since any uncountable closed set in R has the cardi- E-mail addresses: [email protected] , [email protected] , balvarez@ D impa.br (B. Álvarez-Samaniego), [email protected] (W.P. Álvarez- nality of the continuum, it follows that has also the cardinality Samaniego), [email protected] (J. Ortiz-Castro). of the continuum. Moreover, in view of the density of Q in R , the http://dx.doi.org/10.1016/j.joems.2017.02.002 1110-256X/© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) B. Álvarez-Samaniego et al. / Journal of the Egyptian Mathematical Society 25 (2017) 326–330 327 set of Diophantine numbers is also dense in the real line. In addi- Proof. Let f be the function given by tion, D is a set of first category, i.e. it can be written as a countable f : A −→ { 0 , 1 } N union of nowhere dense subsets of R . x −→ y, Now, we would like to prove an interesting result, which will N only be mentioned in this section, related to the fact that the prop- where y = (y n ) n ∈ N ∈ { 0 , 1 } is defined by erty of being a Cantor set is preserved by homeomorphisms when , = , the homeomorphic image of its domain is a closed subset of R . 1 if x2 n +1 1 y n = 0 , if x 2 n = 1 , Proposition 1.1. Let A, B ⊂ R . Suppose that f : A −→ B is a homeo- morphism and B is a closed subset of R . If C ⊂ A is a Cantor set, then for all n ∈ N . From the definition of function f , one gets that f is N N f ( C ) ⊂ B is also a Cantor set. bijective. Then, | A | = | { 0 , 1 } | . Since c = | R | = | { 0 , 1 } | , we con- clude that A has the cardinality of the continuum. Proof. From the fact that C ⊂ A is a perfect set, we get that C is closed subset of R . Moreover, since f −1 is continuous, we have that Using (2.1) , we define the set ( f −1 ) −1 (C) = f (C) is a closed set in B , and since B is a closed sub- + ∞ x − set of R , it follows that f ( C ) is closed in R . Now, we claim that = n 1 = ( ) ∈ . S : x xn n ∈ N A (2.2) ⊂ ∈ ε > ∈ 10 n! f(C) (f(C)) . In fact, let y f(C) and 0. So, there is x C such n =1 that y = f (x ) . Since f is continuous at x , there exists δ > 0 such that The following lemma will be used in the proof of Proposition 2.1 . = ( ) ∈ {− , , } for all z ∈ A, | z − x | < δ ⇒ | f (z) − f (x ) | < ε. (1.2) Lemma 2.2. Let z zi i ∈ Z + be a sequence such that zi 1 0 1 for all i ∈ Z + . If ∈ = , ∈ ⊂ < Considering that x C C there exists z0 C A such that 0 + ∞ | z − x | < δ, and by (1.2) we deduce that | f (z ) − f (x ) | < ε. Fur- z 0 0 i = 0 , = i ! thermore, the injectivity of f implies that f(x) f(z0 ). Then, f(z0 ) 10 i =1 ∈ ( B ( y , ε) ࢨ{ y }) ∩ f ( C ). In consequence, y ∈ ( f ( C )) . Hence, f ( C ) is a = ∈ Z + perfect set in R . On the other hand, since C = ∅ , we see that then zi 0 for all i . f ( C ) = ∅ . Finally, we will show that int ( f (C)) = ∅ . We suppose, by Proof. Since contradiction, that there exists y ∈ int( f ( C )). Then, there is r > 0 + ∞ ( − , + ) ⊂ ( ) ( − , + ) such that y r y r f C . Since y r y r is a connected z 1 z i −1 − = , set, we have that f ((y − r, y + r)) ⊂ C ⊂ A is also a connected i ! 10 = 10 set. Let us take u, v ∈ f −1 ((y − r, y + r)) such that u < v . Then, i 2 ( , ) ⊂ −1 (( − , + )) ⊂ = ∅ u v f y r y r C. Thus, int(C) , which is a contra- we see that diction. Hence, int ( f (C)) = ∅ . Since f ( C ) is a nonempty perfect and + ∞ + ∞ + ∞ z z | z | 1 nowhere dense set in R , we conclude that f ( C ) is a Cantor set. 1 = i ≤ i ≤ 10 10 i ! 10 i ! 10 i ! i =2 i =2 i =2 It is worth mentioning that “every uncountable G δ or F σ set + ∞ 1 in a Polish space contains a homeomorphic copy of the Cantor 1 2 1 < = 10 = . space” [4]. In addition, Alexandroff [5] showed that every uncount- i − 1 = 10 1 90 able Borel-measurable set contains a perfect set. By using these i 2! 10 last facts, one can also obtain some of the results of this paper. | | < 1 < = Thus, z1 9 1. Hence, we conclude that z1 0. However, in Section 2 , we will mainly proceed in a different way, Now, we suppose, by induction, that for n ∈ Z + such that n ≥ by constructing an uncountable perfect and nowhere dense subset = ∈ { , , . , − } 2, we have that zk 0 for k 1 2 n 1 . From the fact that of the Liouville numbers. It deserves remark that Bendixson’s The- + ∞ z z orem, which states that every closed subset of the real line can be − n = i , represented as a disjoint union of a perfect set and a countable set, 10 n ! 10 i ! i = n +1 is used in the proofs of the main results given in Sections 3 and 4 . we get This paper is organized as follows. In Section 2 , the existence of a Cantor set in the set of Liouville numbers is proved. Section 3 is + ∞ + ∞ + ∞ z z | z | 1 devoted to show the existence of a Cantor set inside the set of n = i ≤ i ≤ n ! i ! i ! i ! 10 10 10 10 Diophantine real numbers.
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