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Questions tagged [cantor-set]

For questions concerning the Cantor set, which consists of those real numbers in $ [0,1]$ that remain after repeatedly removing the open middle third of every interval; it contains those numbers which may be written in ternary without using 1. Also, for questions about other topological spaces that are homeomorphic to the Cantor set.

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How do the authors prove that “The relative complement of the Cantor set in $[0,1]$ is dense in $[0,1]$”?

In my textbook Introduction to Set Theory by Hrbacek and Jech, the authors first construct Cantor set: Next they prove The relative complement of the Cantor set in $[0,1]$ is dense in ...
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How to derive closed formulas of Cantor set?

The Cantor set $\mathcal{C}$ is defined as follows: $$\mathcal{C}:=\bigcap_{n=0}^{\infty}C_n$$ where $C_0=[0,1]$ and $C_{n+1} = \dfrac{C_n}{3} \bigcup\left(\dfrac{2}{3}+\dfrac{C_n}{3}\right)$. From ...
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How can I write set of all positive rational numbers

How to write the set of all rational numbers?? From Cantor's matrix we can get the positive ones. Should I add corresponding negative values and a 0?
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Why is the complement of any perfect totally disconnected subset of $\mathbb{R}$ a countable union of disjoint intervals?

The cantor set $C$ is obtained by repeatedly removing the middle $1/3$, starting from the interval $[0,1]$. Since the number of intervals removed in each step of construction is finite, $[0,1] \...
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Space filling curve

Let $C$ be the Cantor set. Then the Cantor function $f:C \to [0,1]$ can be extended to $F:[0,1]\to [0,1]$ linearly as the end points of an removed interval takes the same value. For example $f(1/3)=f(...
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Properties of Cantor set

$[0,1]$ is not homeomorphic to $[0,1]×[0,1]$ but $C$ is homeomorphic to $C \times C$ where $C$ is the Cantor set. I know both the proof. I am asking which property of $C$ is the reason of this ...
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Prove Cantor set is discontinuous at each point

I think my definition of continuous must be wrong, or I am doing something wrong in answering this question. Can someone tell me what is wrong in my proof? where $F$ is the characteristic function, ...
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Minimal dynamical systems in $2^{\mathbb N}$

If we have $\Delta$ a finite set (For simplicity we can just assume it's $2$) and we are looking at $\Delta^\Bbb N$, we can look at this set as a dynamical system with respect to the action: $T((a_n))...
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Proving the continuity of the Cantor Function

Consider the Cantor Set $C=\{0,1\}^{\omega}$, that is, the space of all sequences $(b_1,b_2,...)$ with each $b_i\in\{0,1\}$. Define $g:C\rightarrow[0,1]$ by $$g(b_1,b_2,...)=\sum\limits_{i=1}^{\infty}\...
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Building a Cantor set with positive Lebesgue Measure [duplicate]

Changing the lengths of the intervals excluded during the construction of the ternary Cantor set, show that is possible to build a compact, totally disconnected and perfect set (a Cantor set) with ...
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Locating the subsets where the binary expansion is $0,b_10b_20b_30b_40…$ [closed]

I know that the expansion of decimal numbers is written in the following way $\sum_\limits{i=1}^{\infty} a_i\frac{1}{10^i}$ but on base 2 the binary expansion of decimal numbers is $\sum_\limits{i=1}^{...
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Nested sequence of half open intervals with non-empty intersection

Consider the sets $B_{2^n}^k = [\frac{k}{2^n},\frac{k+1}{2^n})$ with $n \in \mathbb{N}$ and $k \in \mathbb{Z}$. Now we pick a sequence $(k_n)_{n \in \mathbb{N}}$ such that we get a nested sequence $B_{...
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Prove that the piecewise function involving Cantor sets is measurable

Note: $\Psi: \mathcal{Z} \to \mathcal{Z}$, where $\mathcal{Z}$ is the system of finite, disjoint, closed intervals in $[0,1]$ and $\Psi(\dot{\bigcup}_{j=1}^{J}[a_{j},b_{j}]):=\dot{\bigcup}_{...
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1answer
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Neutral element for a Cantor Set.

On Wiki I found the following statement: $T_L$ and $T_R$ together with function composition forms a monoid. I am able to prove the associativity of the composition operation, but what will be a ...
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Koch curve from Cantor sets (paradox)

The Koch curve is normally constructed by taking a line segment, replacing the middle third with two copies of itself forming legs in an equilateral triangle, and then repeating this recursively for ...
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Prove that the Cantor set is homeomorphic to $(X,\mathscr T)$.

For each $n\in \mathbb N$, let $X_n=\{0,2\}$ and let $\mathscr T_n $ be the discrete topology on $X_n$. Let $X=\prod_{n=1}^\infty X_n,$ and $\mathscr T$ be the product topology on $X$. Prove that the ...
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Approximation of Cantor function by piecewise constant function in $L^1$

Let $c(x)$ be Cantor function. How can we prove that constant $\frac{1}{2}$ gives the best approximation in $L^1$ metric? Let $ h(x)= \begin{cases} \frac14 \quad\text{for}\quad x\in[0,\frac13]\\ \...
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“Funny Integral” over the Cantor Set

I was thinking about integrals and how one might generalize them to be able to integrate over fractals rather than just over intervals. For example, consider the cantor set $C$. Let us assume that $$\...
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1answer
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A dense measure $0$ $G_\delta$ subset of the Fat Cantor set?

The fat Cantor set is a nowhere dense subset of $\mathbb{R}$ with positive Lebesgue measure. My question is, does there exist a $G_\delta$ set dense in the fat Cantor set with Lebesgue measure $0$? ...
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A perfect nowhere dense set which intersects every open set with positive measure?

A perfect set is a closed set with no isolated points. A nowhere dense set is a set whose closure has empty interior. My question is, what is an example of a nonempty perfect nowhere dense subset of ...
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$\{0,1\}$-valued continuous functions on the Cantor set

Let $P$ be the ternary Cantor set. I need to determine all continuous functions $f:P\to \mathbb{F}_2$ where $\mathbb{F}_2=\{0,1\}$ is the finite field with two elements. We know that $P$ is ...
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1answer
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The inverse image of a measurable set under a measurable function is measurable?

I have a confusion with measurable functions. I just saw that the statement "The inverse image of a measurable set under a measurable function is measurable" is false with counter-example the function ...
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Can a set of positive measure and its complement both have empty interior? [duplicate]

This might be silly, but I am not sure: Does there exist a Lebesgue measurable subset $E \subseteq (0,1)$ such that $E$ and $(0,1) \setminus E$ both have positive Lebesgue measure. $E$ and $(0,1) \...
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Non empty intersection of cantor set with any open interval is uncountable set

1) Let $C$ denotes the cantor set and $x\in C$ then for any $r>0$, the intersection of $(x-r, x+r)$ with $C$ is uncountable set. true or false? I want to prove that any non empty open set in $C$ ...
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Problems with Cantor's diagonal argument and uncountable infinity

Cantor's diagonal argument: As a starter I got 2 problems with it (which hopefully can be solved "for dummies") First: I don't get this: Why doesn't Cantor's diagonal argument also apply to natural ...
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1answer
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Middle Fifths Cantor Set is Borel and Has Measure =?

I've been working on the following problem; I think I have the answer, I would just like to confirm that there are no gaps in my logic. Consider the "middle fifths" Cantor set $$\mathscr{C}=\left\...
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Prove the Cantor set has no interior.

This is not intended to ask for the proof, but just for you to please check my proof. Thank you. Definition of the Cantor set $C$: $C = C_0 \cap C_1 \cap \cdots \cap C_i \cap \cdots$ where \...
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4answers
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Transition from countable to uncountable

The Cantor set $\mathcal C$ is created by iteratively deleting the open middle third from a set of line segments. Let $\mathcal C_n$ be the set after n iteration. The common definition of Cantor set ...
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Show the following 3 spaces are homeomorphic.

I want to show the following 3 spaces are homeomorphic to one another. $\{0,1\}^{\omega}$, i.e., $\{0,1\}^{\mathbb{N}}$, the set of infinite sequences of 0's and 1's, in the dictionary order topology....
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Is $c( \mathrm{inf} \{ x : c(x) \ge y \}) = y$, for Cantor function $c : [0,1] \to [0,1]$?

Let $c : [0,1] \to [0,1]$ the Cantor function, which is constructed by the following procedure. Convert the input into base 3. If the converted input contains 1, replace every digit after the first ...
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What is the topological dimension of the Cantor set?

For school, I write a kind of ‘paper’ about fractals. I’m currently writing about the definition of fractals, but there’s something I don’t understand. A shape is a fractal, when its Hausdorff ...
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A restricted Cantor Set and Lebesgue measure

This is a problem in Page 85 of Lebesgue Integration on Euclidean Space by Frank Jones. First, the writer constructed a special Cantor Set as following: Choose and positive numbers $l_k$ such that $...
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Evaluating $\int_{C_{\alpha}} f dm$, where $C_{\alpha} \subset [0,1]$ is a generalized Cantor set, $\alpha \in (0,1]$?

Let $f(x) = 3x^2$, I am trying to evaluate $\int_{C_{\alpha}} f dm$, where $C_{\alpha} \subset [0,1]$ is a generalized Cantor set, $\alpha \in (0,1]$. Every continuous function is a measurable ...
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Are countably infinite unions limits?

I was working with the Cantor Set today and its construction prompted me to think about infinite unions/intersections a little more than usual. What exactly does an infinite union/intersection ...
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2answers
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Prove that any Cantor-like set is uncountable.

I have known that there exists some Cantor-like sets with positive measure, and wondering if I can prove that any any Cantor-like set is uncountable by construting or just proving the existence of a ...
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1answer
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Why if 1 occurs somewhere in ternary expansion then that number is not belong to Cantor Set?

I had read some articles about Cantor set and ternary expansion. I come across method of Construction of Cantor set by removing middle third and taking infinite intersection of that set. and there are ...
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Is $f:\mathbb R\to\mathbb R$ concave everywhere if, within each interval on the domain, $\exists$ a sub-interval s.t. $f$ is concave locally?

Let $f:\mathbb R^n\to\mathbb R$ Given that $\forall$ open ball $\mathcal B\subseteq\mathbb R^n$, $\exists$ a subset $\mathcal B_1\subseteq\mathcal B$ s.t. $f$ is locally concave on $\mathcal B_1$, ...
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Singular function that is Holder for all $\alpha<1$

I am asking for an example of a singular continuous function that is Holder for all $\alpha<1$. We know that such function cannot be Lipschitz, otherwise it is absolutely continuous. We also know ...
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1answer
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Cantor set to show that the Borel measure is not complete

I would like to use the Cantor set to illustrate the fact that the Borel measure is not complete. To do so, I saw different sources (wiki and math.stackexchange, if I understood well) using the ...
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minimal equivalence relations free action of a group G on X

Proposition: Let $R$ be an equivalence relation on the space $X$, if the equivalence relation $R$ is minimal, then the only subsets of $X$ which are both closed, open and $R$-invariant are $X$ and the ...
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Cantor set whose measure is 1/10

I’m trying to compute a sequence ‘$p$’ such that $m(C(p))=1/10$. where ‘$m$’ is ‘measure’ and ‘$C$’ is ‘Cantor set’ This is my approach: at step n, I remove $2^{n-1}$ times $p^n$, so the total subset ...
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What does a hyperreal version of the Cantor Set look like?

I would like to construct a hyperreal version of the Cantor set. Let $X_0$ be the interval $[0,1]$ in the hyperreal line, and for any $n$, let and let $X_{n+1}$ be the set of hyperreal numbers ...
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1answer
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Does $X\cdot\langle2\rangle\cdot\langle3\rangle$ cover $\Bbb Z\left[\frac16\right]\cap(0,\infty)$? Is there a similar method using a Cantor set?

Does the set given by $X\cdot\langle2\rangle\cdot\langle3\rangle$ exactly cover $\Bbb Z\left[\frac16\right]\cap(0,\infty)$? Is there a similar method using a Cantor set? Where $X=\{x\in\Bbb N:x\...
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Confusion with the proof that the Cantor set is closed

I have encountered the definition of Cantor set and its property. Cantor set is constructed by removing the middle third open set of each interval .So each time we get some union of closed sets. As $...
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1answer
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Cantor function restriction is onto.

I was reading the construction of Cantor-Lebesgue function in Ziemer's Modern Real Analysis (Page 124). Apparently, this function maps the Cantor set $C$ onto $[0,1]$. Ziemer gives the following ...
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Show that $f: C_{n} \times C_{n}/2 \to [0,1]$ given by $f(x,y) = x+y$ is surjective. ($C_{n}$ is the $n$-step of construction of the Cantor Set)

Show that there exists two closed sets $A, B$ such that $m(A)=m(B)=0$ but $m(A+B)>0$. My question is very similar to this. The difference is that I am required to use $A = \mathcal{C}$ and $\...
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1answer
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Measure Theory and Cantor Set (couterexample).

Consider $O_{n} = \{y \in \mathbb{R}^{d} \mid \exists x \in E \text{ with } |y - x| < \frac{1}{n}\}$ where $E$ is a mensurable set. (a) Prove that if $E$ is compact, then $m(E) = \lim m(O_{n})$...
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1answer
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Existence of an uncountable, pairwise disjoint collection of dense Borel sets, each of which is not in the algebra of open sets

Let $(X, \tau)$ be Cantor space. That is, $X = \{0,1\}^\omega$ and $\tau$ is the collection of open sets in the product of discrete topologies on $\{0,1\}$. Let $\mathcal{A}(\tau)$ be the algebra ...
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A description of the Cantor set, Sierpinski carpet, and Menger sponge.

The Wikipedia article https://en.wikipedia.org/wiki/Menger_sponge characterizes the successive sets $M_n$ whose intersection is the Menger sponge as follows: $M_0 = [0, 1]^3$ and, for each $n \geq 0$, ...
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1answer
117 views

How to prove fat Cantor set has no isolated point

When we prove that the Cantor set has no isolated point, we can use argument about ternary expansion. I feel we cannot use any similar argument to prove the result of fat Cantor set where the ratio of ...