# 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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### 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 ...
2answers
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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 ...
1answer
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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 ...
0answers
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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 ...
1answer
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### 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 ...
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 ...
1answer
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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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### 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-...
1answer
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### 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 ...
3answers
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### 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 ...
1answer
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### 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 ...
3answers
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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/...
3answers
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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 ...
1answer
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### 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 ...
1answer
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### 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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### 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 ...
1answer
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### 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 ...
1answer
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### 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 ...