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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A fat Cantor set essentially containing a positive measure subset and uncountably many of its translates

Edit: In writing out more details, I realized I needed to make a slight modification to the question. I switched all proper containments from the old version of the question to "essential ...
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Let $K \subset [0,1]$ be the Cantor set. show that $f : [0, 1] \to \mathbb R$… [duplicate]

Let $K \subset [0,1]$ be the Cantor set. show that $f : [0, 1] \to \mathbb R$ given by $$f(x) = \begin{cases} 1, & x \in K \\ 0, & x \in [0, 1]\setminus C \end{cases}$$ ...
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Proving the claim that the Cantor set $C$ is perfect [duplicate]

I am self-learning Real Analysis. Stephen Abbott's book leaves the proof of the fact that the Cantor set $C$ has no isolated points as an exercise. I'd like someone to verify, if my construction is ...
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Bogachev's example: $C+C = [0,2]$ and $C-C = [-1,1]$ where $C$ is the Cantor set [duplicate]

It is known, that the Cantor set is the subset of points of the segment $[0,1]$ allowing ternary expansion $0.a_1a_2\ldots$, where $a_i=0$ or $2,\; i\in {\mathbb N}$, neither of those values ($0$ or $...
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1answer
68 views

Cantor space and cantor set [duplicate]

I am trying to show that that the the cantor set is homeomorphic to cantor space and here is the function defined from $C$ cantor set to $\mathcal{C}$ Cantor space, let $a_{n}$ be $0$ or $2$ And for ...
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Proof that $\sum_{n=1}^{\infty}A_n/3^n$ has the Cantor distribution

Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function, according to Wikipedia. In this question, the answer says that the distribution of $\...
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Partitioning a metric space into Cantor sets

A "Cantor set" is a topological space which is homeomorphic to the standard Cantor set $C$. In my answer to the question Another way for partition of perfect set by user 00GB I pointed out ...
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Prove that $C+C=[0,2]$, where $C$ is the Cantor set.

I'd like someone to verify my sketch proof of the below exercise 3.3.7 from Abbott's, Understanding Analysis. If it's incorrect, could you hint/point at the correct approach to the proof. Thanks! ...
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Proving $μ_F(E) = \int_E F' dλ + μ_F(E∩K)$

The following is an exercise from Bruckner's Real Analysis: Let $F$ be the Cantor function. Show that, for every Borel set $E$, $μ_F(E) = \int_E F' dλ + μ_F(E∩K)$, where $K$ is the Cantor ternary set....
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Why is this function increasing?

$K$ : Cantor set $\phi$ : $K \to [0,1]$, given by $\sum_{k=1}^{\infty} \dfrac{2\epsilon_k}{3^k} \mapsto \sum_{k=1}^{\infty} \dfrac{\epsilon_k}{2^k} \ (\epsilon_k=0,1)$ I hear that this function is ...
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Uncoutable Borel Set of Polish Space Contains Cantor Set: Proof Step

Currently, I am struggling to fully understand the following theorem from Alexander Kechris' book “Classical Descriptive Set Theory”: The last sentence is unclear to me. Why is the homeomorphic copy ...
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Continuity of map from Cantor set to $[0,1]$

$K$ : Cantor set $\phi$ : $K \to [0,1]$, given by $\sum_{k=1}^{\infty} \dfrac{2\epsilon_k}{3^k} \mapsto \sum_{k=1}^{\infty} \dfrac{\epsilon_k}{2^k} \ (\epsilon_k=0,1)$ I want to prove that $\phi$ is ...
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Showing function whose derivative is bounded but not (Riemann) integrable

This is how the function is constructed in my notes: Let $C$ be a Cantor set with $\lambda(C)>0$ and $C = [0,1]\setminus U$ where $U = \bigcup\limits_{k=1}^\infty I_k$ is the disjoint union of ...
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47 views

Consider the set, recursively built, starting from the unit interval and removing the first $\frac{1}{3}$ at each step. Find the similarity dimension. [closed]

My thinking for this question is that it is just a slight variation of the standard Cantor set and will therefore have the same similarity dimension. My logic is that at each new step, the interval ...
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1answer
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Integrability of The Indicator of a Perfect Set

I am having trouble solving the following problem: Let $C$ denote the middle-third cantor set. i.e., $C = \bigcap_{i=1}^{\infty}C_n$ where \begin{align*} C_1 &= \left[0, \frac{1}{3}\right] \cup \...
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Variance of the random variable with Cantor distribution

I'm currently looking through Oliver Knill's book Probability Theory and Stochastic Processes and am trying to understand Knill's proof that the variance of a random variable $X$ with the Cantor ...
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Proposition 20, Chap. 2, in Royden's REAL ANALYSIS, 4th edition: The Cantor-Lebesgue Function

Here is the discussion of the Cantor-Lebesgue function in Sec. 2.7, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: We now define the Cantor-Lebesgue function, a ...
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1answer
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For $f$ the Cantor function there is no function $g$ satisfying $μ_f (E)= \int_E g \ dλ$

The following is an exercise from Bruckner's Real Analysis: Let $f$ be the Cantor function, and let $μ_f$ be the associated Lebesgue–Stieltjes measure. Show that there is no function $g$ satisfying $...
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Construction of a function of a countable set of Cantor sets

The following is an exercise from Bruckner's Real Analysis: I represent only ideas I have for each part and if I am in a right track I'll start fill the gaps and make them rigorous. (a) ${\{f<a}\}=...
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Lower bound of middle third cantor set by using frostman property

I am trying to calculate the lower bound for Hausdorff dimension by using Frostman lemma, if we can show that the Frostman property exist then it will automatically proven Hausdorff measure is nonzero....
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Points in fifth middle Cantor set sends to points in fifth middle Cantor set. by a function.

Let $$f(x)= \left\{ \begin{array}{lcc} 1-x & if & x \in [0,1]- \{\frac{1}{10}, \frac{1}{2}\} \\ \\ \frac{1}{2} & if & x=\frac{1}{10} \\ \\ \...
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Necessary and sufficient conditions for a function of a specific type on the complement of the Cantor set

The following is an exercise from Bruckner's Real Analysis: For (a) right limit and left limit must be equal and equals $f(c_n)$ I think this is necessary and also sufficient. For (b) because each $...
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Existence of a sequence of intervals intersects every point of fat Cantor?

The following is an exercise from Bruckner's Real Analysis: Let $C$ be a Cantor set in $[0, 1]$ of measure $α$ ($0 ≤ α<1$). Does there exist a sequence ${\{J_k}\}$ of intervals with $\sum_{k=1}^∞ ...
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Compactness and Cantor's Criterion [closed]

Cantor's Criterion: If $\left\{C_{k}\right\}$ is a nested sequence of closed bounded, nonempty subsets of $\mathbb{R}^{n}$, then $$ \bigcap_{k=1}^{\infty} C_{k} \neq \emptyset $$ Furthermore, if $\lim ...
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How to prove that the cantor function $F_x$ satisfies the property $F_X(u)=\dfrac{F_X(3^nu)}{2^n}$ for $u\in(0,3^{-n})$, $n\in\mathbb N$?

Let $F_x$ be the Cantor function. How to prove that it satisfies the property $F_X(u)=\dfrac{F_X(3^nu)}{2^n}$ for $u\in(0,3^{-n})$, $n\in\mathbb N$?
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What is the Lebesgue measure of a open interval intersected with a generalized Cantor set with positive Lebesgue measure?

In Folland's book Real Analysis: Modern Techniques and Their Applications, p. 39 has an explanation of how to construct a generalized Cantor set with positive measure. For reference, the construction ...
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The measure of a Cantor Set [duplicate]

So I am trying to argue that the set of points in $[0,1]$ which will not have $4$ in their decimal expansion has measure $0$. So I am thinking to construct a step function such that this set is a null ...
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If $f(x)=x^3-\lambda x$ then $\Big\{ x\in\mathbb R : \Big| \lim_{n\to \infty}f^n(x)\Big|<+\infty \Big\}$ is a Cantor set.

This problem is from An introduction to chaotic dynamical systems Robert L. Devaney I've already prove (a) and (b) but i'm a bit confused about (c). For (c) we ...
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105 views

Show that a number belongs or does not belong to a Cantor set

I have trouble showing that $\frac{10}{31}$ and $\frac{45}{69}$ belong/do not belong to the Cantor set. I tried to do the following $$\sum_{n = x}^\infty \frac{2}{3^n} = \frac{2}{3^x} + \frac{2}{3^{(x ...
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1answer
25 views

Condensation points and Lebesgue-zero measure set

Definition. $x \in \mathbb{R}$ is a condensation point of a subset $A \subseteq \mathbb{R} \iff$ the intersection of every neighbourhood of $x$ with $A$ is uncountable. Can one construct a subset $N \...
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Representation of numbers in Cantor set as ternary expansion - in need of a rigorous proof

Background Facts It is known that every number in the Cantor set $\mathcal{C}$ has a ternary expansion, where $a_k \in \{0,2\}$: $$ x=\sum_{k=1}^{\infty} a_{k} 3^{-k} $$ Cantor set has the ...
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1answer
28 views

Cantor set and evaluating an integral

I'm trying to prove that the following function is Riemann integrable on $[0,1]$: $$g(x) = \begin{cases} 5, &x\in C \\ x, &x\not \in C\end{cases}$$ where $C$ denotes the Cantor set. It's well ...
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Compact Zero-Dimensional, Planar Sets are Contained in Planar Cantor Sets?

It's known that a zero-dimensional, compact metric space $X$ embeds in the Cantor Set: Take a countable, clopen basis $(U_j)$ indexed by $\mathbb{N}$ and construct a map $f: X \rightarrow \lbrace 0, 1 ...
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BV[a,b]$\cap$C[a,b]$\neq$AC[a,b]

Take an $f\in$BV[0,1]$\cap$C[0,1] e.g. the Cantor function. I take the Lebesgue Stiltigies measure of $f$: $$ \mu_f((a,b])=f(b)-f(a). $$ Now I have a finite positive measure, so I can do the Radon-...
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88 views

Is this a probability measure on the cantor set?

$c$ is the Cantor function, $C$ the Cantor set, and $\rho$ the Lebesgue measure We consider $\mu$ as $\mu(A) = \rho(c(A \cap C))$ for each each element of the tribute Is $\mu$ currently defining ...
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67 views

Cantor set intersected with $\mathbf{R}\setminus\mathbf{Q}$

I am trying to construct an example of a perfect set that contains no rational. I see in other places a construction that involves an enueration of rationals, but I thought that construction was a bit ...
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36 views

Is the graph of $f$, $G_{r}(f)$, is connected?

Consider the Cantor set $C$ in $[0,1]$. And a function $f: [0,1] \rightarrow [-1,1]$ defined by $$f(x) = \frac{2(x-a)}{b-a}-1 \text{ if } x \in [a,b] \text{ where }(a,b) \text{ is a contiguous ...
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Prove the set of contiguous interval of Cantor set is countable.

Prove the set of contiguous interval of Cantor set is countable. Remember that means one of the "holes" you cut out when constructing the Cantor set and we know that Cantor set is ...
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Prove there is a sequence $\{x_n\}, x_n $ in Cantor set and not interval contiguous such that $x_n \rightarrow \frac{1}{3}$

Let $C=$ Cantor set. We know by construction of Cantor set that endpoints of contiguous intervals of the cantor set, for example $(\frac{1}{3},\frac{2}{3})$ is a contiguous interval and $\frac{1}{3}, \...
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Properties about Cantor sets.

Let $C$ be the usual Cantor defined $C = \bigcap F_n$ where $F_1 = [0,1]$, $F_2 = [0,\frac{1}{3}]\cup[\frac{2}{3},1]$, $\dots$. I am trying to prove some elementary properties about this set and was ...
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How to show two properties about the Cantor Set

Define $C_0=[0,1]$ and for $n\in\mathbb{N}$, define $$C_n=C_{n-1}\setminus\bigg(\bigcup_{k=0}^{3^{n-1}-1}\bigg(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\bigg)\bigg) $$ Then the Cantor set is defined as $$C=\...
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1answer
73 views

Homeomorphic images of endpoints of the Cantor set.

A Cantor set is a homeomorphic image of the standard ternary Cantor set $T$. Suppose that we have a Cantor set $C$ on the plane. It is well know that $C$ is in fact ambiently homeomorphic to $T$, ...
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continuous map from cantor set to another set

Let $C$ be the Cantor set with metric from $(\mathbb{R},|\cdot|)$ and $P = \prod_{k=1}^\infty \{0,\frac{1}{2^k}\}$ with metric from $(\ell_1, \lVert \cdot \rVert_1).$ Show that there exists a ...
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What does it mean to find something in Set theory? What is formal-language equivalent of “find”?

For example in proof that every real number $x$ has a decimal expression $x=a_0.a_1a_2a_3….$ it says: so we can find $a_1$ between $0$ and $9$ such that... https://math.stackexchange.com/a/2625318/...
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If the set of specific sequences of 0's and 1's is countable then it can be used to count the set of all infinite sequences of 0's and 1's. Right?

I am seeking some help to understand why an infinite sequence of $\{0, 1\}$ is uncountable. While there are similar questions here with detailed answers, I wasn't able to resolve the contradiction ...
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1answer
65 views

How can the Cantor set be uncountable, yet its complement is the union of only countably many disjoint open intervals? [duplicate]

$C$ = Cantor set. $C$ is closed, so $C^c$ is the countable union of disjoint open intervals. So let $$C^c = \bigcup\limits_{\substack{i~\in~\mathbb{N} \\}} A_i $$ where $A_i$ are disjoint open ...
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1answer
39 views

Where does surjectivity fail in a mapping from the unit interval to the Cantor Set?

I'm asked to state why there cannot be a surjective and continuous function $$g: [0,1]\longrightarrow \text{Cantor Set} $$ I know that $g^{-1}$ exists and is continuous & surjective since the ...
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2answers
88 views

Proving $g:P\to\Bbb R, g(x)=3x+2$ is discontinuous

Define $g:P\to\Bbb R,g(x)=3x+2$ where $P$ is the Cantor set. Show that $g$ can't be continuous under Euclidean metric in $\Bbb R$. No information is given about the metric on $P$. I assumed it is the ...
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1answer
37 views

Union of deletion Cantor sets

I know that it is possible to have deletion Cantor set which are of non-zero measure (fat Cantor sets) furthermore it must never of measure one since it would result in a contradiction (as Cantor is ...
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1answer
41 views

Ternary and Binary representations to Prove Cantor set is uncountable (questions)

I am trying to prove that the Cantor set is uncountable, but I am very much a novice in working with different bases of numbers. I've never had to do it in any of my classes until now. Where I'm at is ...

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