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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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How can complement of the Cantor set be the countable union of open intervals?

I studied a long back about Cantor sets. Today while explaining it to one of my friends I got stumped at the complement of the Cantor set. As far as I am aware of, the complement of the Cantor set is ...
Anil Bagchi.'s user avatar
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Almost all projections of the four corner Cantor set have $0$ Lebesgue measure. A mistake in the proof and how to fix it

Let $E$ be the $1/4$ Cantor set, which I define briefly here $$E := \{ 3\sum_{n=1}^{\infty}{\frac{\epsilon_n}{4^n} } \, : \, \epsilon_n = 0 \text{ or } 1\}$$ Or it can be defined in this way Let $\...
Paul's user avatar
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0 answers
26 views

Sarkovskiis theorem and the Cantor set

Can we prove the following as such (with relevance with the Sarkovskiis theorem)? Suppose that $f$ is continuous and that $A_0 , A_1 ,\dots, A_n $ are closed intervals and $f(A_i) \supset A_{i+1}$ ...
Superunknown's user avatar
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2 votes
2 answers
96 views

Looking for an efficient (computerized) way to convert large ternary sequences into their decimal equivalents.

I am doing a project on Cantor Sets for my undergraduate and I need an efficient (computerized) way to convert large ternary expansions such as the following to their decimal equivalent. $$0....
Sidekiq's user avatar
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0 answers
22 views

Alternative proof Box-counting dimension Cantor set

From the Book of Falconer (Fractal Geometry: Mathematical Foundations and Applications, 2nd ed.), we have the following proposition: Proposition 3.2 Let $F \subset \mathbb{R}^n$, then $$ \begin{...
Mikeys00's user avatar
2 votes
2 answers
71 views

Distance in Cantor space

We have $C = 2^\mathbb{N}$ and we identify the Cantor set with the image of $C$ via: $ t\colon C \rightarrow [0,1], \left(x_n\right)_n \mapsto \sum_{n=1}^{\infty}\frac{2x_n}{3^n} $ Now if $x,y\in t(C)$...
strugglingStudent's user avatar
6 votes
5 answers
1k views

Proof by contradiction, that a set of all binary sequences, where "1" cant be twice in a row, is uncountable

I emphasize that I want to prove it by contradiction using cantor diagonal method. $$A = \{ (a_i) \mid \text{$a_i$ all the binary sequence where $1$ doesn't appear twice in a row}\}.$$ So I'm ...
Yarden Tziar's user avatar
0 votes
1 answer
37 views

Understanding Falconer Example 4.2

Presently, I am reading Falconer's book on Geometric Measure theory. In example 4.2 he used mass distribution principle to calculate the lower bound of the Hausdorff dimension of Cantor set $C$. Now ...
Mayank's user avatar
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36 views

Can we construct Cantor set by any integer which is strict bigger than 2.

As we also know, the normal Cantor set is constructed by keeping cutting out the middle 1/3 set. So, can we construct a Cantor set $E$ by keeping cutting out the nth 1/m set, where $1 \leq n \leq m$, ...
xxxg's user avatar
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1 answer
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Can we use the Cantor function to prove that Cantor set is uncountable?

Since the cantor function is an onto function from the cantor set to $[0,1]$, the cantor set must be uncountable. Is this statement right? I think there is a simpler proof for it than what I have seen....
xxxg's user avatar
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2 votes
1 answer
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When is the union of two locally closed subsets locally closed? [closed]

It isn't true in general that the union of two locally closed subsets (i.e., subsets of the form open $\cap$ closed) are locally closed, so is there some standard condition that guarantees it?
mathematrucker's user avatar
1 vote
1 answer
59 views

Example of a continuous measure which is mutually singular wrt Lebesgue measure

We are are asked to find an example of a measure $\lambda$ such that $$f(x)=\int_0^x d\lambda >0 $$ for all $x>0$, $f$ is continuous in $[0,1]$, and $\lambda$ is mutually singular wrt the ...
Marta Sánchez Pavón's user avatar
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63 views

How to prove that the distance $|x-y|$ in $A$ (where $A$ is a subset of $\mathbb{R}$ with positive lebesgue measure) does not belong in the Cantor set

Obviously if their distance is above 1 they don’t, but I’m curious on how to approach this for $|x-y|<1$. A thought is to prove that for this set $A$ there exist two elements $x$ and $y$ whose ...
NoetherBoy 's user avatar
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1 answer
70 views

Applying Zorn's lemma to Cantor set

Let $C_n$ be the set obtained in the $n$-th step for constructing the Cantor set. It is not hard to see that $C_i \subseteq C_j$ if $i>j$ and we know that the Cantor set is $\cap_{n=0}^{\infty} C_n$...
RHspqr's user avatar
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2 votes
1 answer
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Zero Set of the Minimum of Brownian Motions

We know that the zero set of a Brownian motion $(B(t))$, $T:=\{s\in [0,1]:B(s)=0\}$, is almost surely homeomorphic to the Cantor set. I would like to prove that the zero set of the minimum the ...
user1598's user avatar
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0 votes
0 answers
49 views

Subset $V$ Lebesgue measurable, but not Borel measurable

Define the function $g: [0, 1] \rightarrow [0, 1]$ with $$g(y) := \inf\{x \in [0, 1] : f(x) = y\},$$ where $f$ is the Cantor function. Now let $V \subset [0, 1]$ be a set that is not Lebesgue ...
Minerva's user avatar
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0 answers
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Show that the image of a homeomorphism involving the Cantor function has a certain measure

Currently working on the task of showing the following: Let $f$ be the Cantor function and $$g: [0,1] \rightarrow [0,1], x \mapsto \frac{f(x)+x}{2}$$ Then $\lambda(g(C)) = \frac{1}{2}$ where $\lambda$ ...
Zedssad's user avatar
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0 answers
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Show the Lebesgue-measure of the image of the Cantor-Function is equal to 1?

Currently strugging to proof the following: Let f be the Cantor-Function defined as the limit of affine functions (so not the definition with base 3), C the cantor set and $\lambda$ the Lebesgue ...
Zedssad's user avatar
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0 answers
75 views

Function that maps (binary representation of) $x \in [0,1]$ to number given by the “even bits” of $x$

I’m just wondering if there’s a name for the function defined below. Write $x$ uniquely in base 2 (i.e. not ending in an infinite string of ones) as $x = \sum_{k=0}^{\infty}\frac{x_k}{2^k}$. Define $...
MathFrak96's user avatar
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41 views

Show that the union occurring in the definition of cantor-sets is disjoint.

The Cantor set C can also be defined as follows: Let $ A_{0}:=[0,1] $ and $ A_{k}:=\frac{1}{3} A_{k-1} \cup \frac{1}{3}\left(A_{k-1}+2\right) $ for $ k \in \mathbb{N} $. Then $ C=\bigcap_{k=0}^{\infty}...
Euler007's user avatar
  • 132
1 vote
1 answer
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Closure of a disjoint union of open intervals

My question mainly is wether or not the closure of a disjoint union of open intervals is the union of the closure of those intervals, because I cannot think of any counterexample, and if this is the ...
H4z3's user avatar
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0 answers
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Finding the error with this proof that the measure of the cantor set is 0

I'm having some trouble with the following exercise: Prove that the Cantor set $C$ has Lebesgue measure of zero. This is what I did: I will use $mA$ to denote the lebesgue measure of $A$. We can ...
Eduardo Magalhães's user avatar
0 votes
1 answer
78 views

Extend the homeomorphism of Cantor set to a homeomorphism to unit interval.

Let $C_1$ and $C_2$ be any two Cantor sets with different ratio $α$ ($0<α<1$, where $1/3$ is the standard ratio).Show that there exist a function $F:[0,1]\to [0,1]$ with $F$ being continuous ...
lee's user avatar
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1 vote
0 answers
54 views

proof of Cantor-Bernstein Theorem

let $f$, $g$ injective functions $f:E\rightarrow F$ and $g:F\rightarrow E$ consider this set defined as follow $\mathcal F=\left\{C\subset E, g\left(F-f(C)\right)\subseteq E-C\right\}$ i have proved ...
ana nadi lwa3r's user avatar
1 vote
1 answer
106 views

Cantor set and expressing $[0,1)$ as a countable union of disjoint closed intervals

The problem reads as follows: Prove that $[0,1)$ cannot be expressed as a countable union of disjoint closed intervals. While I was able to solve the problem, I am interested in how to solve it ...
giochi's user avatar
  • 498
0 votes
1 answer
79 views

Construct a Cantor-type subset [duplicate]

This problem is a part of an exercise in Zygmund's Real Analysis. It reads: Construct a measurable set $E$ of $[0,1]$ such that for every sub-interval I, both sets $E\cap I$ and $I\setminus E$ have ...
Apple's user avatar
  • 79
0 votes
1 answer
85 views

Confused about the measure of the Cantor Set, and how to reconcile this with there being points not at the endpoints

I know that there are points not at the endpoints of the intervals we have removed in the cantor set, such as $1\over 4$, because we can represent the numbers remaining by all ternary strings using ...
Alexander D.'s user avatar
1 vote
1 answer
102 views

Why is the Cantor Set not a subset of $\mathbb{Q}$?

I am looking for an example of a specific element of the Cantor set which isn't a rational number, and how it comes about in the set when constructing it, to understand why the Cantor set isn't a ...
Princess Mia's user avatar
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3 votes
1 answer
56 views

Showing that a set similar to Cantor set has complement with measure one

Consider the Cantor set $C$. Define a new "Cantor" set which we will call $C_\alpha$ for $\alpha \in (0,1)$. We construct $C_\alpha$ by letting $C_0 = [0,1]$, $C_1 = C_0$ but with an open ...
Grigor Hakobyan's user avatar
0 votes
1 answer
144 views

questions about the Cantor set is Compact

enter image description here In Rudin's Chapter 2, Section 2.44, the Cantor set is described as a union of infinitely many intervals. It is true that we can find an infinite open cover that covers ...
ssds's user avatar
  • 3
-2 votes
1 answer
62 views

Range of a function on subintervals [duplicate]

Prove or disprove the existence of a function $f:[0,1] \rightarrow[0,1]$ with the following property: for any interval $\,(a,b)\subset[0,1]\,$ with $\,a\!<\!b,f\big((a,b)\big)\!=\![0,1]\,.$ It ...
BlizzardWalker's user avatar
0 votes
1 answer
70 views

How many "disconnections" are there in the Cantor set?

Question How many "disconnections" are there in the Cantor set? Countably many, or uncountably? What do I mean? I've deduced that an uncountable, totally disconnected set can be made from a ...
it's a hire car baby's user avatar
0 votes
0 answers
51 views

Properties of Cantor-like function

I recently encountered the following function, defined in terms of the (standard, middle-thirds) Cantor set $\mathcal{C}$, on the domain $[0,1]$: $$f(x) = \min\{y \in \mathcal{C}: y \geq x\}$$ Since ...
Pavel S.'s user avatar
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0 answers
93 views

Determining whether a natural number has no "1"s in its ternary expansion, without direct calculation

I am considering a problem concerning Cantor set. Here is my quesion: Let $N\in\mathbb{N}$. Is there any way other than direct calculation to determine whether $N$ has no "1" as digits in ...
tanjia's user avatar
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0 votes
0 answers
42 views

An exercise about the Cantor measure

I got stuck on exercise 11 in chapter 6 of Michael Taylor's Measure Theory and Integration. Let $Z=\prod_{k=1}^{+\infty} \Big\{ 0,\,2\Big\}$ and $G: Z \to [0,1]$ given by $$ G(a)=\sum_{k=1}^{+\infty}...
Matteo Aldovardi's user avatar
1 vote
2 answers
230 views

Has this "thinner" Cantor set been defined and studied before?

The Cantor set is defined by starting with the closed interval $[0,1]$ and repeatedly taking "bites" of open intervals in the middle, like $(1/3,2/3)$. However, I have thought of something ...
user107952's user avatar
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1 vote
1 answer
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To prove that the Cantor function maps the end points of the intervals removed to a same point

For the function f in 1.6D i.e. $$x=0.b_1b_2....(3)$$ then $$f(x) = y= 0.a_1a_2.....(2)$$ where $a_i=b_i/2$ Here (2),(3) represents the binary and ternary expansions of $x$ respectively. Show that if $...
Lakshmi Priya's user avatar
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0 answers
62 views

Does anyone have a nicer closed formula for the Cantor function?

I just spent this morning working on a closed formula for the Cantor function. Here is what I got: For $x\in[0,1],$ $$c(x)=\sum_{m=1}^\infty\left(\lfloor(3^mt(x)\rfloor-3\lfloor3^{m-1}t(x)\rfloor\...
Miles Gould's user avatar
0 votes
2 answers
98 views

Bijection between Cantor set and binary sequences; Hunter

I am reading John K. Hunter's lecture notes, specifically chapter 5, section 5.5., about the Cantor set. I have some questions about the proof of the theorem stating that the Cantor set has the same ...
psie's user avatar
  • 777
2 votes
0 answers
66 views

Self-homeomorphisms of the Cantor set

Let $C$ be the Cantor set. I was wondering about what we know about the topological group $Homeo(C)$ of self-homeomorphisms of $C$. I am looking for example for a classification of its elements (I ...
Nicolas Guès's user avatar
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0 answers
58 views

Does this work to prove that the cantor set lacks isolated points?

I came up with this idea to show that the Cantor set lacks isolated points (i.e. every point is a limit point) but I'm not sure if it works. Let $C$ be the Cantor middle third set. My goal is to show ...
roundsquare's user avatar
  • 1,507
0 votes
1 answer
143 views

Prove that union of intersections of nested balls in $R^k$ is a perfect set

Exercise: Let $S_n$ for $n \geq 1$ be a finite union of disjoint closed balls in $R^k$ of radius at most $2^{-n}$ such that $S_{n+1} \subset S_n$ and $S_{n+1}$ has at least 2 balls inside each ball of ...
Stanislav Veklenko's user avatar
2 votes
0 answers
51 views

An increasing, continuous function $f$ such that $f'=0$ a.e. outside an arbitrary exceptional uncountable set with measure 0.

Recall that the famous Cantor-Lebesgue function $\phi (x)$ is a continuous, increasing surjection from $[0,1]$ to $[0,1]$ with the property that $\phi (0)=0$, $\phi(1)=1$ and derivative equal to $0$ ...
VShaw's user avatar
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0 answers
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Is the collapsed Cantor set homeomorphic to the interval $[0,1]$?

If I take the Cantor ternary set constructed on the interval $[0,1]$ and collapse the spaces between the points of the Cantor set, do I obtain an interval again? My intuition tells me that it should ...
Sarah April's user avatar
1 vote
1 answer
240 views

Constructing a Homeomorphism with the Cantor Set

This question is about a blog post by Terrence Tao showing that the real line $\mathbb{R}$ cannot be expressed as a disjoint union of countably many closed intervals. There are many proofs for this ...
Nick A.'s user avatar
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0 answers
74 views

The Cantor set is uncountable even though it is a closed set with outer meaure $0.$ ("Measure, Integration & Real Analysis" by Sheldon Axler. )

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler. The author wrote the following sentence. Sentence 1: Now we can use the Cantor function to show that the Cantor set ...
tchappy ha's user avatar
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0 votes
1 answer
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Given any $k\in\Bbb{N}$ there exists $k$ consecutive composite integers such that reciprocal of each composite integer is a Cantor point then $k=1$

Conjecture: Given any $k\in\Bbb{N}$ there exists $k$ consecutive composite integers with the property that reciprocal of each composite integer is Cantor points then $k=1$. $x$ is a Cantor point iff ...
Sourav Ghosh's user avatar
2 votes
1 answer
169 views

Irrational numbers in the Cantor set. ("Measure, Integration & Real Analysis" by Sheldon Axler.)

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler. 2.74 Definition Cantor set The Cantor set $C$ is $[0,1]\setminus (\bigcup_{n=1}^{\infty} G_n)$, where $G_1=(\frac{1}...
tchappy ha's user avatar
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2 votes
0 answers
156 views

Understanding why the Cantor function maps some Lebesgue measurable subset to a non measurable one

I'm going to present the following problem and a solution given in an older colloquium in Measure Theory in hope of understanding a statement in the final conclusion. $\boldsymbol{\underline{\text{...
PinkyWay's user avatar
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1 vote
0 answers
25 views

Confusion over dense level sets of the Takagi function

I was told that the Takagi function, $T:[0,1]\rightarrow \mathbb{R}$, is continuous and has uncountable dense level sets in $[0,1]$. This has confused me for the following reason: Suppose $L$ is a ...
JDoe2's user avatar
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