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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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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 claims that the "answer" is ...
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How to calculate the weight of the Cantor dust?

(or "What is wrong in cardinality × size of the elements of the Cantor set?") Supposing the "standard" Cantor set definition, $$ \mathcal{C} = \bigcap_{k=1}^\infty C_k $$ where $|C_k|=2^k$, the ...
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Why Cantor set is not definied 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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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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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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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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1answer
61 views

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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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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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....