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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 to prove that rational numbers are countable to someone who won't accept the visual proof? [on hold]

I am trying to prove to someone that the rational numbers are countable by using the proof shown on the website linked here: https://www.homeschoolmath.net/teaching/rational-numbers-countable.php. ...
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24 views

Is the length of each interval in the Cantor set itself a member of the Cantor set?

Considering the Cantor set, which can be covered with a sequence of intervals defined as the following: $$\{[a_{di}, b_{di}], d=1, 2, ..., \infty, i=1, 2, ..., n_d\}$$ where $d$ indexes the "depth" ...
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23 views

Continuous extension from Cantor set to unit interval

I have the following assignment: Let $C \subset [0, 1]$ be the Cantor set, $x ∈ C$ satisfying $$x = \sum_{n = 1} ^ \infty \frac{a_n}{3^n} \; ,$$ and $\varphi: C \to [0,1]$ a function defined as $$\...
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Irrational numbers in the set of real numbers: Cantor set

I am writing my thesis on quasi-periodic oscillations, which are signals containing two frequencies (let's leave it by that for now) with an incommensurable (irrational) ratio. However, I am a trained ...
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1answer
33 views

Cantor function sends ternary expansions to binary expansions

Let $F_0(x)=x$ The Cantor set in $[0,1]$, obtained by removing middle thirds. Then we have a Cantor function $F(\sum_{k=1}^{\infty} a_{k}3^{-k})=\sum_{k=1}^{\infty}\frac{a_k}{2}2^{-k}$ for $a_k \in \...
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43 views

Cantor Set Homeomorphic to Inverse Limit Space

For each positive integer $n$ let $X_n = \left\{1, 2, 3, ..., 2^n \right\}$ with the discrete topology and let $f_n : X_{n+1} → X_n$ be the function defined by: $f_n(i) = i$ for $1 ≤ i ≤ 2^n$ $f_n(i)...
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Does every null set have a superset which is an $F_{\sigma}$ null set?

Let $A$ be a Lebesgue null set in $\mathbb R$. Can we find a set $B$ with the following properties: 1) $A \subset B$ 2) $B$ has measure $0$ 3) $B$ is an $F_{\sigma}$ set (i.e. a countable union of ...
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1answer
33 views

Cantor-like Set not of measure zero

I have a short question, but that keeps me stuck for a couple of days. Let's start saying that Cantor Set is defined as: $$C:=\left\{x\in\mathbb{R}|x=\sum_{n\in\mathbb{N}}\frac{\alpha_n}{3^n}, \...
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56 views

Cantor Set Homeomorphic to Itself

This question is partly taken from Hatcher’s notes on point-set topology. Given standard $1/3$ Cantor set on $[0,1]$, we need to show that a function $f_S:C \to C$ is a homeomorphism, where $f_S$ is ...
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Cantor Set is Homeomorphic to the Countable Product of {0, 1}

In Part of proof of homeomorphism from Cantor set ot infinite product of {0,1} it is shown that the pre-image of sets of the form $U(j,a)= \left\{ (a_n)_{n=1}^{\infty} ∈ \left\{ 0,1 \right\}^N :a_j=...
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Formula for canonical basis of the Cantor set

Is there a natural formula for the canonical clopen subsets of the middle-thirds Cantor set $C$ which is based on elements $\sigma\in 2^{<\omega}$: $B(\varnothing)=C$ $B(\langle 0\rangle)=[0,1/3]...
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Define function and image of fat Cantor set

This construction I found it paper published in $1965$ I think. Here is the way that defined. Let $I=[0,1]$ and define a Cantor set as follows. $C_1$ obtained from $I$ by taking the open interval ...
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To prove that Cantor function is not absolute continuous, do we need to take countable infinite number of subintervals?

This highly upvoted answer claims that we can show that the Cantor function is not absolute continuous by taking a finite number of subintervals. However, one comment right below the answer claims ...
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Why is the Cantor set not defined by a limit?

The Cantor set, as showed in books and Wikipedia, is defined in terms of $C_k$, the finite Cantor set of level $k$: $$ \mathcal{C} = \bigcap_{k=1}^\infty C_k $$ But after intersection only the "last"...
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Using union of countably infinite sets, I tried to prove that set of all real numbers in [0,1) is countable

Cantor's diagonal method shows that the set $S=\{x\in \Bbb R|x \in [0,1)\}$ is uncountably infinite, because there is no bijection between the set $S$ and the set of natural number $\Bbb N$. I came ...
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prove that the ternary cantor set is compact and a perfect set.

prove that the ternary cantor set is compact and a perfect set. My trial: I know that I should prove that it is closed and bounded, for proving that it is closed because finite union of closed sets (...
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Cardinality of Sub(A) is smaller than ${\aleph_0}$ or equal $2^{\aleph_0}$?

Let $\mathbb{A}=(A,\mathcal{F}^\mathbb{A})$ be a countable algebra. I need to prove that $|Sub(\mathbb{A})|\leqslant\aleph_0$ or $|Sub(\mathbb{A})|=2^{\aleph_0},$ where $Sub(\mathbb{A})$ is the ...
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1answer
32 views

Explicit homeomorphism between product of Cantor sets onto the Cantor set

I want to find an explicit homeomorphism $\varphi: C\times C \longrightarrow C$ where $C$ denotes the Cantor set. The hint is to use the base $3$ expansion of the elements of the Cantor set. My two ...
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Let $X$ be the topological space formed by $K$ and an isolated point $p \notin K$. Show that $X \cong X^2$ but $X \ncong X^n$ [closed]

K is the usual Cantor set. I was studying and I came across this problem, but I could not solve it.
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Homeomorphism from the real numbers to the real numbers with restriction to the Cantor set.

Let $K$ be the Cantor set and $C \subset R$ be a non empty compact set with no isolated points and empty interior. Prove that it exists a homeomorphism $f:R \longrightarrow R$ such that $f(K)= C$.
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Convolution square of the Cantor set

For $0\leq d\leq 1$, let $\eta_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$ (for some normalization); recall that it is translation-invariant. Motivation for what follows: Up to ...
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1answer
62 views

Quotient space $[0,1]/C$ is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,…\right\}$, $C$ denotes Cantor set. [closed]

How to prove quotient space $[0,1]/C$, where $C$ denotes Cantor set, is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,...\right\}$?
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Continuous map from subset of $\mathcal{C}$ (Cantor) to non-measurable set.

I've come across the following proposition and its proof. Given $f(x)=x+V(x)$, where $V(x)$ is the Cantor-Vitali function on $[0,1]$: $f(x)$ is an homeomorphism from $[0,1]$ to $[0,2]$. $\...
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Integral of cantor function

Let $\phi$ denote the cantor ternary function and $\mu = \mu_{\phi}$ the associated measure. Let $\lambda$ denote the lebesgue measure. Compute the following integrals. $\int_{[0,1]} \phi(t)d\mu (t)$ ...
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Alternate definition of middle-$\alpha$ Cantor set using affine transformations

In most books, the middle-third Cantor set is described algorithmically or with a picture (i.e., showing the first few steps of removing middle-thirds of intervals). Here, I am trying to approach the ...
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Cantor distribution extended

Let $A_{m,n}$ be the $m$-th removed interval (from left to right) in step $n$ of constructing the Cantor set $C$. For example $A_{1,2} = (1/9,2/9)$, $A_{2,2} = (3/9,4/9)$, $A_{3,2} = (7/9,8/9)$. ...
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Formal representation of the numbers of the Cantor set.

I have already proved that: Proposition 1. Let $x\in[0,1)$, then exists $\{c_k\}_{k\in\mathbb{N}}\subseteq\mathbb{N}$, with $0\le c_k\le p-1$ such that $$x=\sum_{k=1}^{+\infty}\frac{c_k}{p^k},$$ ...
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1answer
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Asking about a hint: constructing a cantor-like set

This problem is a part of an exercise in Stein's Real Analysis. It reads: Construct a measurable set $E \subset [0, 1]$ such that for any non-empty open sub-interval $I$ in $[0, 1]$, both sets $E \...
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1answer
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G is the Cantor set, $X$ is a countable set of G hence G\X is dense

Exercise: Let $G$ be a Cantor set. a) Prove that $G$ has a countable dense subset. b)If $X$ is a countable subset of $G$ then $G\setminus X$ is dense in $G$ a) It is know that the Cantor ...
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The cantor set $K$ contains no intervals

Let $K$ the Cantor set. I already proved the following properties Properties.\begin{equation} \begin{split} (1)&\quad|K|=|\mathbb{R}|\\ (2)&\quad\lambda(K)=0,\text{where}\;\lambda\;\text{is ...
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Formal construction of the Cantor set $K$ and determination of $[0,1]\setminus K.$

Let $J_{0,1}:=[0,1]$. Step 1. We remove the central open interval $I_{0,1}=\big(\frac{1}{3},\frac{2}{3}\big)$. We denote with $J_{1,1}:=\big[0,\frac{1}{3}\big]$ and with $J_{1,2}:=\big[\frac{2}{3},...
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How to prove this closed formula for Cantor set?

Let $C_0=[0,1]$ and $C_{n+1} = \dfrac{C_n}{3} \bigcup\left(\dfrac{2}{3}+\dfrac{C_n}{3}\right)$. Theorem: $$C_n=\bigcap_{m=0}^{n}\bigcup_{k=0}^{\left\lfloor \frac{3^{m}}{2}\right\rfloor}\left[\...
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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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43 views

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 $$\...