# 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 space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure?

Question : Does there exist a space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure ? My attempt : The problem is I have no idea whether it is true or false. If I were to ...
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### Can the quotient space obtained by partitioning the closed interval into Cantor sets be Hausdorff?

In response to this question Can the Interval be Covered by Disjoint Cantor Sets? it was pointed out that the answer is, Yes: see Theorem 1.14 of Paul Bankston and Richard J. McGovern, Topological ...
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### An Inequality related to the Cantor Set $\mathscr C \subset [0,1]$

Let $\mathscr C\subset\mathbb R$ be the Cantor set on the interval $[0,1]$. Let $x\in \mathscr C$, and $0 < r < 1$ such that $$\frac{2}{3^k} < r \le \frac{2}{3^{k-1}}$$ for some positive ...
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### Cantor-like functions for $\xi\neq \frac{1}{3}$

Let $\xi \in (0,\frac{1}{2})$. Let $C_\xi$ be the perfect symmetric set built by iterating the transformation $$[0,1] \to [0,\xi]\cup [1-\xi, 1].$$ The set $C_\frac{1}{3}$ would then correspond to the ...
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### Kunen exercise IV.4.13 (4): Topological version of effective AC

I am dealing with Kunen's The Foundations of mathematics exercise IV.4.13 (4): Let $X$ denote the Cantor set. Prove if $S\subset X\times X$ is open, then there is an $F\subset S$ such that $F$ is the ...
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### Does there exist any continuous function $f:\mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R}\setminus \mathbb{Q}) = \mathbb{R}$?

My attempt to the problem is first I intend to make a perfect nowhere dense set of irrational numbers of measure $0$ like cantor set(and that I guess we could do)..now make such type of function like ...
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### Constructive proof of representation for open balls in Cantor space

The Cantor set is the set of all numbers that can be written in the following form: $$\mathfrak{C} = \sum_{i = 1}^{\infty} \frac{a_i}{3^i}$$ for an infinite sequence $a_i$ such that $a_i = 0$ or $2$ ...
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### Bound on variation $|F(x)-F(y)|$ where $F$ is Cantor's function

Let $F$ be Cantor's function. Prove that there exists some $C>0$ s.t for all $x,y\in[0,1]$ $$|F(x)-F(y)|\leq C|x-y|^\alpha$$ Where $\alpha=\log_3 2$. My direction of thought was to use the ...
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### How than I show that this map is continuous?

I have the following problem: Consider the set $M=\{0,2\}^\mathbb{N}$ and the map $f:M\rightarrow C$ where C is the Cantor set such that $f(a)=\sum_{n=1}^\infty \frac{a_n}{3^n}$ and show that f is ...
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### Cantor function integral.

I'm trying to solve the following integral for a problem about fractals involving Cantor set: $$\mathcal{I}=\int_{0}^{1}C\left(\sqrt{1-x^2}\right)dx$$ Where $C(x)$ denotes the Cantor ternary function. ...
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### Evaluating the integral of the Cantor function

Define the Cantor set $\mathcal{C}:=[0,1]\setminus\bigcup_{n=1}^{\infty}G_n$, where $G_1=(\frac{1}{3}, \frac{2}{3})$ and $G_n$ for $n>1$ is the union of the middle-third open intervals in the ...
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### Properties of fat Cantor set

Let $C$ be a fat Cantor set and $\mathbb{Q}$ is set of rationals. Q) Is it true that closure of intersection of $C$ and $\mathbb{Q}$ is $C$ that is $$\overline{ C\cap \mathbb{Q}}=C?$$ Q) If the above ...
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### Every open interval of $\mathbb{R}$ contains either infinitely many or no elements in the Cantor set

I am thinking about a proof of the following statement: "Every open interval of $\mathbb{R}$ contains either infinitely many or no elements in the Cantor set" and this is what I have thought:...
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### Is the Cantor set $\mathcal{C} = \bigcap_{k=0}^{\infty} C_k$ the set of all the endpoints of the closed intervals of the $C_k$'s?

Let $\mathcal{C}$ be the middle-third's Cantor set. For each integer $k \geq 0$, let $C_k \subset [0,1]$ denote the union of disjoint closed intervals obtained at the $k$th stage of the construction ...
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### Rescaling set that intersects each perfect set

Suppose $D\subset\Bbb R$ such that $D\cap P\neq\emptyset$ for each nonempty perfect set $P\subset\Bbb R.$ Notice that $D$ need not to be a Bernstein set. Clearly, $D$ intersects each perfect set in ...
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### Can we take a Cantor set as a witness of inner regularity?

Let $X$ be a compact Polish space without isolated points and $\mu$ be a finite atomless Borel measure on $X$. Then, for all $0 < \varepsilon < \mu(X)$, is there a subset $A$ of $X$ which is ...
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### Minimal sets, Perfect sets, Exceptional sets and Foliations

The first image is taken from Geometric Theory of Foliations, Book by A. Lins Neto and César Camacho, Chapter 3, Page 53. The second image is taken from Geometry, Dynamics And Topology Of Foliations: ...
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### 'Simple' reference for dynamically defined cantor sets

I have recently been made to believe, that a better understanding of dynamically defined cantor sets will be useful to compute Hausdorff dimension of a specific class of sets I'm interested in. ...
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### How to view the middle third Cantor set as a fixed point via it's self-similarity

The definition of middle third cantor set is given in this link" https://en.wikipedia.org/wiki/Cantor_set and we need to use Hausdorff metric, the definition of the Hausdorff metric is given in ...
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### Does the Cantor Set with Irrational Endpoints Contain Rationals?

Let $[a,b]$ be a nonempty interval with irrational endpoints. Choose distinct irrational points $p,q\in(a,b)$ Remove the subinterval $(p,q)$ from the initial interval $[a,b]$ Repeat the process ...
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### Any nonempty totally disconnected, perfect compact metric space is homeomorphic to Cantor set - proof explanation.

I am reading the proof of the theorem 30.3 from General Topology by Stephen Willard: Any two totally disconnected, perfect compact metric spaces are homeomorphic. The corollary from this theorem is ...
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