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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Cantor set with filled "gaps"; Is my solution correct?

Let's modify Cantor set. Let $f_{\gamma}$ be a function that maps interval $[x,x+d]$ to $[x,x+d\gamma]\cup[x+d(1-\gamma),x+d\gamma]$, i.e. it removes middle part of length $d(1-2\gamma)$ (we consider $...
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Under what condition a meagre set has measure $0 $?

Under what condition a meagre set has measure $0$ ? I know the existence of fat cantor set or cantor like set which is meagre but has positive lebesgue measure. My question :We can say a ____$\text{...
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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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A question about partial inverses of a map from the Cantor set

Let $X = \{0,1\}^\mathbb{N}$ with the metric $d(x,y) = 2^{-\text{min}\{i|x_i\neq y_i\}}$. It is known that $X$ is homeomorphic to the standard Cantor set. Define a map $f:X\to \mathbb{R}/\mathbb{Z}$ $$...
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Symmetry, homeomorphic sets, and the Cardinality of Even and Whole number sets being equal

Very similar questions to this have been answered on the following threads, but they have not answered this particular question about the symmetry of the relation used for calculating the cardinality. ...
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Are Cantor-like sets disjoint for $\xi,\eta$ with no common power?

Let $\xi,\eta \in (0,\frac{1}{2})$. Let $C_\xi$ (and analogously for $C_\eta$) be the perfect symmetric set built by iterating the transformation $$[0,1] \to [0,\xi]\cup [1-\xi, 1].$$ Will the sets $...
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Geometric series don’t coincide

Let $j\in \{1,2\}$. For each $j$, assume that $(a^j_n)_{n\ge0}\subset\{0,1\}^\infty$ is a sequence with $a^j_n\in\{1,0\}$ for each $n$. Choose $\xi_1,\xi_2\in (0,\frac{1}{2})$. Let $ d^i_n$ be the ...
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Ternary sequence

Consider $p\in\mathbb{N}$, $p\ge 2$ we know that fixed $x\in[0,1)$ exists a sequence $\{a_k\}\subseteq\mathbb{N}$ such that for each $k\in\mathbb{N} $ we have $0\le a_k\le p-1$ and$$x=\sum_{k=1}^{\...
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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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Cantor-like Sets are Nowhere dense.

I attach the Cantor-like set construction here: Let $\alpha \in (0, 1)$. Define the "middle-$\alpha$" of a closed interval by $$ M_\alpha([a, b]) = (-\alpha, \alpha) \frac{b - a}{2} + \frac{...
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How does the distance of a function from cantor set look like?

While reading this result (which claims continuity but actually proves uniform continuity), I wondered how do such functions (on $\mathbb R$) look like? For a singleton set $\{a\}$, it would be $|x-a|$...
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Are points in the cantor set limits of endpoints

If $x\in C$ the cantor set and $S$ is the set of all endpoints in the iterative construction (i.e. $C=\cap C_i$ and $C_i= [a_1^i,b_1^i]\cup...\cup[a^i_{2^i},b_{2^i}^i]$ then $S =\cup_i\cup_k \{a^i_k,b^...
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For any Cantor set $C \subset R^n$ there exists a submanifold $M \subset R^{n+1}$ such that $M \cap (R^n \times \{0\}) = C$

Show that for any Cantor set $C \subset R^n$ there exists a submanifold $M \subset R^{n+1}$ such that $M \cap (R^n \times \{0\}) = C$. I think $M$ should not be transverse to $R^n \times \{0\},$ ...
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Sequences, Series and Cantor Set (show it is uncountable using series given) - Beginner level

Let $(c_n)$ be a sequence with $c_i ∈ ${$0, 1$} for all $i$. The series from $n=0$ to infinity, $2c_n/3^{n+1}$. This series converges. Now, for each $(c_n)$, the series is an element of $C$. I need to ...
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Looking for a set like the fat Cantor set?

Problem: Assume $0 < \epsilon < 1$ and $\mu_{\mathcal L}$ is the Lebesgue measure. Find a measurable set $A \subset [0, 1]$ such that the closure of $A$ is $[0, 1]$ and $\mu_{\mathcal L}(A) = \...
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Induction-based proofs to show the Cantor Set is Perfect

$\def\C{\mathcal{C}}$ $\def\N{\mathbf{N}}$ $\def\R{\mathbf{R}}$ Let $C_0 := [0,1]$. Having already defined $C_n$, we define $$C_{n+1} := \frac{C_n}{3} \cup \left( \frac{2}{3} + \frac{C_n}{3} \right)$$ ...
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Does $\mathbb{Q} \times $ Cantor set have a complete sequence of $\sigma$-discrete closed covers?

Does the space $\mathbb{Q} \times \mathcal{C}$ possess a complete sequence of $\sigma$-discrete closed covers? I am interested in this question, because if answered positively, the Theorem 1 in the ...
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1 answer
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If two points are chosen at random from the Cantor set, what is the average distance between them?

I've so far figured that I need to look at different cases for if the points are within the lower or upper third of the Cantor set, but am confused on how to proceed beyond that to take an average for ...
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Continuity of piecewise function involving rationals/irrationals and Cantor set

I'm struggling to determine the continuity of the following function: $$ f(x)=\begin{cases}0 \quad \text{if $x \in \mathbb{Q}\cap D,$}\\ x^3 \quad \text{if $x \notin \mathbb{Q}\cap D$;} \end{cases}$$ ...
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Construction of a nowhere dense perfect set

I have to construct for every $\varepsilon > 0$ a nowhere dense perfect set $\mathit K \subseteq [0,1]$ with $\lambda(K) > 1 - \varepsilon$. A hint is to do it similar as constructing the Cantor ...
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Expression of closed intervals appeared in the $m$-th stage of construction of the Cantor set

I am reading this paper and have a question related to the following line: Quote: Let $D'$ be the collection of all closed discs in the $xy$ plane whose diameters are the intervals in the $x$-axis $$\...
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7 votes
1 answer
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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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4 answers
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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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1 vote
1 answer
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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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4 votes
1 answer
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Ring homomorphisms between $C([0,1], \mathbb{R})$ and $C(\text{Cantor set},\mathbb{R})$

Let $K \subseteq [0,1]$ be the Cantor set. $C([0,1], \mathbb{R})$ be the ring of continuous functions from $[0,1]$ to $\mathbb{R}$. $C(K,\mathbb{R})$ be the ring of continuous functions from $K$ to $\...
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Need clarification on definition of cantor distribution

In https://en.wikipedia.org/wiki/Cantor_distribution it says it's a probability distribution with CDF the cantor function $C(x)$. I'm a bit confused on what the probability distribution actually is. ...
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Cantor function satisfies for $x\in[{0,{1\over 3}]}$ $F(x)={1\over 2}{F(2x)}$

Define Cantor function $F:[0,1]\to [0,1]$ to be $F(x):=\lim_{n\to \infty}{\int}_{n=0}^{\infty}g_n(t)dt$ where $g_n(t) = \begin{cases} \big({{3\over 2}})^n, & \text{if $t\in C_n$ } \\ 0, & \...
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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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3 votes
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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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1 answer
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Two equivalent definitions of the Cantor function

I know two definitions of the Cantor function $c: [0,1] \to [0,1]$. $$ c(x) = \begin{cases} \sum_{n=1}^{\infty} \frac{a_{n}}{2^{n}}, \; x \in C\\ \sup_{y \leq x, \; y \in C} c(y), \; x \notin C \end{...
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Cantor space has a binary tree structure?

I know that every point in the Cantor set has a binary tree representation. i.e., there exist closed intervals, $\{ I_{n}^j \}_{n\in \mathbb{N}, 1\leq j\leq 2^n}$, such that $$ \mathcal{C}= \bigcap_{...
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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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Is it true that if $A$ and $B$ are nowhere dense subsets of $[0,1]$ then the Minkowski sum $A+B=\{a+b:\ a\in A, b\in B\}$ is nowhere dense in $[0,2]$?

This is one of my weird "isolated problems" that I just think up by myself. Question $1:\ $ Is it true that if $A$ and $B$ are nowhere dense subsets of $[0,1]$ then the Minkowski sum $A+B = \...
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5 votes
1 answer
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A question about the Cantor Lebesgue function

I am reading some lecture notes on the Cantor Function. Here is the construction that notes used. The notes used the Cantor set to construct a function. Here the Cantor set is defined to be $\{O_n \}$ ...
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-1 votes
1 answer
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Open covers of Cantor set

The Cantor (1/3) set $C$ is constructed as follows: $ C_0=[0, 1] $, $ C_1=[0, 2/3] \cup [2/3, 1] $, $ C_2=[0, 1/9] \cup [2/9, 1/3] \cup [2/3, 7/9] \cup [8/9, 1] $, and so on. Then $ C= \bigcap_{i=1}^{\...
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1 vote
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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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2 votes
1 answer
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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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2 votes
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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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showing that a point is not an end point of any interval of the Cantor set

I know that for example the point $\frac{1}{4}$ is in the Cantor set. But how do I show that is not an endpoint of any interval in the Cantor set? My attempt: At step $n$, we remove the open interval $...
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1 vote
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Denjoy's Counterexample, Semi-Conjugation, Irrational Rotation, $C^1$ Diffeomorphism and Cantor Set in $\mathbb{R}$

Introduction to the Geometry of Foliations, Part A. Gilbert Hector, Ulrich Hirsch. Pages 73, 74. Why is the map $J$ everywhere continuous from the right and continuous from the left for points not ...
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