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.
0
votes
1answer
17 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 ...
3
votes
1answer
33 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 ...
0
votes
1answer
41 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?
0
votes
1answer
56 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] \...
0
votes
1answer
53 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(...
0
votes
1answer
45 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 ...
0
votes
1answer
33 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
votes
1answer
101 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))...
1
vote
2answers
47 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}\...
0
votes
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 ...
1
vote
2answers
34 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}^{...
1
vote
2answers
27 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_{...
0
votes
3answers
80 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
99 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
votes
0answers
61 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
votes
0answers
33 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
88 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
$$\...
0
votes
1answer
38 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
votes
1answer
73 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
votes
0answers
34 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
votes
1answer
146 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 ...
4
votes
2answers
316 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) \...
0
votes
0answers
14 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 $C$ ...
1
vote
2answers
100 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
92 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
votes
0answers
90 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
votes
4answers
44 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 ...
-2
votes
1answer
57 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....
1
vote
1answer
39 views
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 ...
-1
votes
1answer
35 views
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 ...
0
votes
0answers
26 views
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 $...
0
votes
0answers
18 views
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 ...
0
votes
3answers
30 views
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 ...
2
votes
2answers
79 views
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 ...
0
votes
1answer
36 views
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 ...
0
votes
1answer
56 views
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 $\mathcal B_1$,
...
0
votes
0answers
23 views
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 ...
2
votes
1answer
136 views
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 ...
0
votes
0answers
28 views
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 ...
-1
votes
1answer
57 views
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 ...
0
votes
1answer
56 views
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 ...
2
votes
1answer
40 views
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\...
0
votes
3answers
133 views
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 $...
2
votes
1answer
21 views
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 ...
1
vote
1answer
29 views
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 $\...
3
votes
1answer
43 views
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 $m(E) = \lim m(O_{n})$...
3
votes
1answer
93 views
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 ...
1
vote
0answers
44 views
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 $n \geq 0$, ...
1
vote
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 ...
