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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0answers
32 views
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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1answer
39 views
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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1answer
84 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
27 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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0answers
77 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
25 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
36 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 ...
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1answer
54 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
78 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
37 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
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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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0answers
19 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
47 views
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
31 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
55 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 ...
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3answers
60 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
56 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
101 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: $$C_n=\bigcap_{m=0}^{n}\bigcup_{k=0}^{\left\lfloor \frac{3^{m}}{2}\right\rfloor}\left[\...
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1answer
20 views
How do the authors prove that “The relative complement of the Cantor set in $[0,1]$ is dense in $[0,1]$”?
In my textbook Introduction to Set Theory by Hrbacek and Jech, the authors first construct Cantor set:
Next they prove The relative complement of the Cantor set in $[0,1]$ is dense in ...
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2answers
61 views
How to derive closed formulas of Cantor set?
The Cantor set $\mathcal{C}$ is defined as follows: $$\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
44 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
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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1answer
81 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
48 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
59 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
110 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
58 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
23 views
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
36 views
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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2answers
29 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
87 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}_{...
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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
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2answers
106 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 ...
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0answers
75 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 ...
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0answers
38 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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1answer
98 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
$$\...
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1answer
51 views
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$?
...
2
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1answer
78 views
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 ...
5
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0answers
39 views
$\{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 ...
5
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1answer
205 views
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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2answers
346 views
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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0answers
19 views
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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2answers
162 views
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 ...
1
vote
1answer
124 views
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\...
3
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0answers
154 views
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
\...
0
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4answers
48 views
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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1answer
65 views
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....
