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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51 views
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.
...
0
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1answer
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" ...
1
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1answer
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
$$\...
1
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0answers
39 views
Irrational numbers in the set of real numbers: Cantor set
I am writing my thesis on quasi-periodic oscillations, which are signals containing two frequencies (let's leave it by that for now) with an incommensurable (irrational) ratio. However, I am a trained ...
1
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1answer
33 views
Cantor function sends ternary expansions to binary expansions
Let $F_0(x)=x$
The Cantor set in $[0,1]$, obtained by removing middle thirds.
Then we have a Cantor function $F(\sum_{k=1}^{\infty} a_{k}3^{-k})=\sum_{k=1}^{\infty}\frac{a_k}{2}2^{-k}$ for $a_k \in \...
2
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1answer
43 views
Cantor Set Homeomorphic to Inverse Limit Space
For each positive integer $n$ let $X_n = \left\{1, 2, 3, ..., 2^n \right\}$ with the discrete topology and let $f_n : X_{n+1} → X_n$ be the function defined by:
$f_n(i) = i$ for $1 ≤ i ≤ 2^n$
$f_n(i)...
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2answers
56 views
Does every null set have a superset which is an $F_{\sigma}$ null set?
Let $A$ be a Lebesgue null set in $\mathbb R$. Can we find a set $B$ with the following properties:
1) $A \subset B$
2) $B$ has measure $0$
3) $B$ is an $F_{\sigma}$ set (i.e. a countable union of ...
1
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1answer
33 views
Cantor-like Set not of measure zero
I have a short question, but that keeps me stuck for a couple of days.
Let's start saying that Cantor Set is defined as:
$$C:=\left\{x\in\mathbb{R}|x=\sum_{n\in\mathbb{N}}\frac{\alpha_n}{3^n}, \...
0
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1answer
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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1answer
30 views
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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1answer
21 views
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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1answer
48 views
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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2answers
60 views
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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1answer
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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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1answer
107 views
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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1answer
49 views
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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1answer
92 views
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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1answer
37 views
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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0answers
34 views
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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0answers
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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 ...
2
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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 $[0,1]/\left\{0,1,\frac12,\frac13,...\right\}$?
2
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1answer
89 views
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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0answers
74 views
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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0answers
15 views
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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0answers
20 views
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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0answers
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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
42 views
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
59 views
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 ...
2
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3answers
70 views
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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0answers
62 views
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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2answers
114 views
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: $$C_n=\bigcap_{m=0}^{n}\bigcup_{k=0}^{\left\lfloor \frac{3^{m}}{2}\right\rfloor}\left[\...
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1answer
25 views
How do the authors prove that “The relative complement of the Cantor set in $[0,1]$ is dense in $[0,1]$”?
In my textbook Introduction to Set Theory by Hrbacek and Jech, the authors first construct Cantor set:
Next they prove The relative complement of the Cantor set in $[0,1]$ is dense in ...
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2answers
78 views
How to derive closed formulas of Cantor set?
The Cantor set $\mathcal{C}$ is defined as follows: $$\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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1answer
46 views
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
65 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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1answer
98 views
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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1answer
52 views
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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1answer
92 views
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, ...
2
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1answer
122 views
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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2answers
75 views
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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0answers
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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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2answers
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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}^{...
1
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2answers
33 views
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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3answers
94 views
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}_{...
1
vote
1answer
31 views
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 ...
5
votes
2answers
126 views
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 ...
3
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0answers
117 views
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 ...
0
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0answers
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]\\
\...
8
votes
1answer
108 views
“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
$$\...
