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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Can I determine if the result of a pairing function is the from the inverse of a given pair?
For example, say you had the following results:
f(a, b) = c
f(b, a) = d
Is there a pairing function that would allow for determining that c is sort of the "inverse" of d without de-pairing ...
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What are the least number of sub-intervals to cover the genralized cantor set in terms of its definition and length of sub-intervals covering it? [closed]
Suppose we redefine the generalized cantor set as the "$m/n^{\text{th}}$ cantor set", where $m$ is a positive integer and $n$ is positive odd integer. We remove $m$ number of $1/n^{\text{th}}...
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Continuity of the indicator of Cantor set.
Consider the following problem which appears in N.L.Carothers.
Let $g:\mathbb {R\to R}$ be a function defined by $g(x)=1$ if $x\in \Delta$ and $g(x)=0$ otherwise,then find the points of continuity of $...
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1answer
43 views
Why is this set $\mathbf{\Pi}^0_\xi$?
$\newcommand{\C}{\mathcal C}$
Let $\C$ denote the Cantor space and let $U\subseteq \C\times\C$ be $\C$-universal for $\mathbf{\Sigma}^0_\xi(\C)$, for some $1\leq\xi<\omega_1$, meaning that $U\in\...
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2answers
71 views
Solving for complex exponent
Background (though not necessary for the mechanics of what I'm asking): According to Lapidus' $\textit{Fractal Geometry, Complex Dimensions and Zeta Functions}$, the Cantor String consists of lengths $...
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41 views
Does removing open intervals with equal spacing give a Cantor-like set
For the almost Mathieu operator, which can be described by the following SL$(2,R)$ matrix, dubbed the transfer matrix,
$$
T(E, n;\omega) = \begin{pmatrix} E - 2\lambda \cos(2\pi\omega n) & -1 \\ 1 ...
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25 views
Cantor theorem use
I decided to use Cantor's theorem to demonstrate by contradiction but I'm not entirely sure if what I did is correct (since I had a lot of trouble understanding it). Can you tell me if what I did is ...
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1answer
59 views
Topological difference between the compact interval $I$ and the Cantor set
There is an homeomorphism between the Cantor set $X = 2^\omega$ (with the product topology) and the Cantor ternary set $\mathcal{C}=[0,1] \smallsetminus \bigcup_{n=0}^\infty \bigcup_{k=0}^{3^n-1} \...
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Properties of the Cantor set constructed by removing just the middle point
Context:
I am currently doing some exercises on the middle $\lambda$ Cantor set $C_\lambda$ (construction is similar to the usual Cantor set, but we remove the middle $\lambda$ proportion of the ...
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Let $C$ be the Cantor set. Prove $g(x)=\left\{\begin{array}{ll}1 & x\in C \\ 0 & x\notin C\end{array}\right.$ not continuous at any $c\in C$.
Is this proof sound? Also it feels like this proof is unnecessarily confusing, but I'm not sure how to improve it.
Proof:
Let $c \in C$ and $\epsilon = 1$. Let $\delta > 0$. Since $\displaystyle \...
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1answer
27 views
Cantor Ternary Set Problem, Ternary Expansions
I'm reading Lebesgue Integration on Euclidean Space by Frank Jones and I'm stuck on this problem, of page 41, problem 15.
Let $C$ the Cantor Ternary Set.
Let $G_{1}=(\frac{1}{3},\frac{2}{3})$, $G_{2}...
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29 views
Hausdorff measure of Cantor set
I am trying to find a set $A\subset\mathbb{R}$ with Hausdorff dimension $\log2/\log3=:s$ but has $H^s(A)=\infty$.
I suspect this is the Cantor set, but im struggling to show that it has Hausdorff ...
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42 views
False reasoning with cantor set and probability.
Hello I hope you are doing great.
I would like to know your opinion regarding this reasoning.
Assume you that you put a point in the $n$-th iteration of the ternary Cantor set. This set is formed ...
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1answer
17 views
Showing the map from the middle $\lambda$ cantor set to the $\nu$ cantor set is $\gamma$-hölder continuous
Let $C_\lambda$ and $C_\nu$ be the middle $\lambda$ and $\nu$ cantor sets, respectively. I want to show the map $\Pi_{\lambda,\nu}:C_\lambda\rightarrow{C_\nu}$ is $\gamma$-hölder continuous, with $\...
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1answer
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Contradiction involving the Cantor Set and Baire Category Theorem
I was given a "proof" that contradicts the Baire Category Theorem and I can't figure out why it doesn't work.
Let $C$ be the Cantor set in $[0,1]$. Then $[0,1]\setminus C$ is a disjoint union of ...
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Measure of meagre/nowhere dense sets in the Cantor Space
$\newcommand{\N}{\mathbb{N}}$
$\newcommand{\Z}{\mathbb{Z}}$
Let $E = \{0, 1\}^\Z$, with the usual product topology. Let $A$ be a meagre set of $E$, and let $\mu$ be a measure on $E$ with the ...
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Approximating the Characteristic Function of a Cantor Type Set with Continuous Functions in Stromberg's Book
On page $276$ of his book Stromberg mentions that $\xi_{P}$ (where $P\subset[0,1]$ is a Cantor type set) can be written as the pointwise limit of a sequence of contiuous functions on $[0,1]$.
He ...
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1answer
19 views
Details on a proof of the uncountability of the Cantor Set
So, I'm trying to prove that the Cantor Set is uncountable using some compactness argument, but I have some doubts about particular details. My proof attempt is as follows:
"Proof": Let $P$ denote ...
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2answers
35 views
Open neighborhoods in the set of $K=\prod_1^{\infty}\{0,1\}$
The problem given to me goes as follows:
Define $K=\prod_1^{\infty}\{0,1\}$, in the product topology. Let $S=s_n$ be a sequence of nonnegative real numbers such that $\sum_1^{\infty}s_i=1$. Define a ...
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25 views
Prove that the following surjection of the Cantor set onto unit square is continuous.
Assume that $C$ is a Cantor Set defined in the following way:
$$C = \left\{\sum_{n = 1}^{\infty}\frac{a_n}{3^n} : a_i \in \{0 ,2\}\right\} \subset [0, 1] =: I$$
with induced topology.
I want to show ...
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Proving That The Cantor Set is Uncountable Using Base-3
I've been given the following explanation of why the Cantor set is uncountable using base-3 (shortened):
In base-3 we write some arbitrary number $x \in [0,1]$ as
$$x = 0.b_1b_2b_3..._3 = \frac{...
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1answer
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Why does Cantor's diagonalization not disprove the countability of rational numbers?
Say we enumerate the list of rational numbers in the way given in the standard proof of rational numbers being countable (the link of the proof is given below). Then we take all of the numbers from ...
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3answers
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How to prove that Cantor ternary set has no intervals using ternary expansions?
Found this proof on this site: Cantor's Set Has No Intervals but I don't understand a part of it.
Here is the proof:
Suppose that $x,y\in C$, where
$$x=\sum_{k\ge 1}\frac{a_k}{3^k}\quad\text{...
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1answer
58 views
Find the average of $P$ defined on the Fat Cantor set
Suppose we have $P:A\to[0,1]$, where $A$ is the fat cantor set denoted as $C$.
We produce $C$ by removing $1/4$ of $[0,1]$ around mid-point $1/2$
$$C_{1}=[0,3/8]\cup [5/8,1]$$
$$C_{1,1}=[0,3/8] \ \ ...
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2answers
93 views
Find the average of a function defined on a Fat Cantor Set
Suppose we have $P:A\to[0,1]$, where $A$ is the fat cantor set denoted as $C$.
We produce $C$ by removing $1/4$ of $[0,1]$ around mid-point $1/2$
$$C_{1}=[0,3/8]\cup [5/8,1]$$
$$C_{1,1}=[0,3/8] \ \ ...
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0answers
56 views
Can we define a measure, for arbitrary uncountable null sets, that gives the average of a function between the infimum and supremum of range?
Suppose $P:A\to\mathbb{R}$ where $A$ is arbitrary uncountable, null and a subset of $\mathbb{R}$
According to comments under here, unless an uncountable null-set is shift-invariant, you won't likely ...
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1answer
331 views
Prove $(X,d)$ is a metric space where $|X| < |\mathbb{R}|$, then $X$ is zero-dimensional.
Can I please receive help with part (ii) of this problem? I have no clue how to solve it. It is from Munkree's topology book. Thank you!
$\def\R{{\mathbb R}}
\def\N{{\mathbb N}}$
A topological ...
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1answer
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Cantor Function
Hello I have an exercise to do but I’m stuck on a few questions. The statement is as follows :
Consider the Cantor function f: [0,1] —> [0,1] With $f(x) = \sum_{j=1}^{N(x)} \frac{1}{2^j} 1(x_j \ge 1)...
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Solving a limit sum involving cantor sets
Let $P:C\to \mathbb{R}$, where $C$ is the cantor set and $P$ is continuous. If $C_n$ is the sequence of numerators from the values of endpoints from the defined intervals of iteration $k$
$0, 1, 2, 3,...
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1answer
124 views
Average of function defined on cantor set
Suppose we have $P: A\cap[0,1]\to\mathbb{R}$, where $A$ is the Cantor set. I want to define and find the average of $P$ to give a result between the infimum and supremum of $P$'s range.
Obviously we ...
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Does there exist any direct proof that the Cantor set is countably dense homogeneous?
A separable space $X$ is said to be countably dense homogeneous (CDH) if for any two countable dense subsets $A$ and $B$ of $X$ there exists a homeomorphism $f:X\longrightarrow X$ such that $f(A)=B.$
...
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1answer
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Is there any visual proof that rationals in cantor set are dense in cantor set?
I was thinking about how we can get a feel that rationals in Cantor set are dense in Cantor set. Is there any way to put this thing in a visual way? It is quite easy to think for irrationals in the ...
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1answer
17 views
Show that for every interval $(a,b)$, the intersection of C (Cantor set) with $(a,b)$ is empty or uncountable
My try:
Case I
Let $(a,b)$ such that for any subinterval of $C = \bigcap_{m=0}^{\infty}F_m$, $[\frac{n}{3m},\frac{n+1}{3m}]<(a,b)<[\frac{n+2}{3m},\frac{n+3}{3m}]$ for some $n \in \mathbb{N}$
...
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Unbounded Cantor like set.
Consider the interval $[0,\infty)$,if we construct a subset of this unbounded closed interval as follows,
$A=\bigcup_{n\in \mathbb N}$$(n+C)$, where $C$ denotes Cantor set and $n+C$ is the translation ...
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1answer
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Prove that any open interval $(a, b)$ that intersects $\bf C$ contains some subinterval of some $F_m$.
Prove that any open interval $(a, b)$ that intersects C (Cantor's set) contains some
subinterval of some $F_m$. Then conclude that $(a, b)$ also intersects the complement
from C.
My try:
In other ...
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1answer
38 views
Cantor's problems: the number $\xi =_{(3)} 0.222000222000…$ in $\mathbb{C}$ is a rational number $\frac{p}{q}$. Find $p$ and $q$.
Let C be the Cantor's ternary set.
a) If $\xi =_{(3)} 0.02002000200002...$ is an element of C write which are the subintervals of $F_0, F_1, F_2, F_3, F_4$ and $F_5$ to which it belongs $\xi$.
b) ...
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An identity needed to construct cantor function
So I'm constructing the cantor-lebegue function this way, given $x\in [0,1]$ let $a_k$ be the coefficients in the ternary expansion for $x$ and $K\in\mathbb{N}$ the first coordinate such that $a_K=1$, ...
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What is the geometrical meaning of “approximate differentiability” and how much is it different from “differentiability almost everywhere”?
A function $f:D\subset\mathbb R^n\to\mathbb R^m$ is said to be differentiable a.e. (almost everywhere) if the exceptional set $E:=\{p\in D:f\text{ is not differentiable at }p\}$ have Lebesgue measure ...
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2answers
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Decide which of the following indicated functions $f$ are homeomorphisms and which are not. Show clearly why or why not.
My partial solution is wrong, can someone please help me with this? I know we need to show bijection and continuous but not sure how.
Let $C$ denote the Cantor set, $\{0,1\}^{\mathbb{N}}$ with the ...
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1answer
60 views
Split perfect and Cantor sets
Let $C$ be a Cantor ternary set. We can write it as union of $\mathfrak{c}$ many disjoint Cantor sets as follows:
$$ C\times C= \bigcup_{x\in C} (\{x\}\times C)$$ and clearly $\{x\}\times C$ is ...
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1answer
32 views
Is there a closed nowhere dense set with no isolated points from the left?
I want to find an example (or the insight that there is no such set) of closed nowhere set with no isolated points from the left. I tried to write down some constructions of which I wasn't able to ...
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2answers
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Cantor space with prefix metric
Consider the metric space $\{$ $a_0a_1a_2....$ $:$ $a_i=0,1$ $\}$
with the metric $d(x,y)=min(I: a_i\neq b_i)$ and 0 otherwise.
Edit: I meant to say $d(x,y)=\frac{1}{2^{min(I:a_I\neq b_I)}}$ when $x\...
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2answers
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Homomorphisms and the Cantor's set
Let C be a Cantor set and let U be a subset of C which is both open and closed. Then U is a Cantor set.
I don't know how to prove that...
Any hints would be appreciated.
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2answers
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The Cantor set and its equivalents.
Let $(\{0,1 \}^ {\mathbb {N}},d)$ be a metric space, where $$d((a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}})=\sum_{n=1}^{\infty}{\dfrac{|x_{n}-y_{n}|}{2^{n}}},\phantom{a}\forall (a_{n})_{n\in\...
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1answer
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Topological properties on Cantor set
Show that the cantor set with the usual topology on $R$ is a normal , regular subspace.
It seems to me that is true as the cantor set is subset in $[0,1] \subset{R}$ which is a metric space . So it ...
0
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1answer
59 views
Finding the measure of the generalized Cantor set.
I am working on the following problem from Royden's Real Analysis book:
Let $F$ be the subset of [0, 1] constructed in the same manner as the Cantor set except that each of the intervals removed at ...
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Embedding Cantor set into a Surface
I'm looking for a result that generalized the following theorem to closed surface.
Theorem: (All Cantor set in the plane are tame) Suppose $C$ is a Cantor set and $f:C \to \mathbb{R}^2$ is an ...
3
votes
1answer
110 views
What are the topological properties of the Cantor set if a point is added at the center of every missing segment?
What are the topological properties of the Cantor set if a point is added at the center of every empty segment?
Suppose that when the segment $\left(\frac13,\frac23\right)$ is deleted from $[0,1]$, a ...
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votes
1answer
80 views
$C+ \frac 1 n C=[0,1+1/n]$ conjecture
Let $n$ be a positive integer. Let $C \subseteq[0,1]$ be the ternary Cantor set. I have made the conjecture $C+\frac 1 nC=[0,1+1/n]$ and am wondering if it is true or not.
It is obviously true when $...
2
votes
1answer
153 views
Showing “directly” that a fat Cantor set contains a non-measurable subset
Let $L$ be the Volterra set made by removing the middle open interval of length $5^{-\nu}$ at each step. In the limit, this will yield a perfect compact set of positive Lebesgue measure, i.e. a "fat ...
