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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Sequence in $C_c(\mathbb{R}^{d})$

i want to find a sequence $(f_i)_{i\in \mathbb{N}}$ in $C_c(\mathbb{R}^{d})$ such that $f_i = \chi[0,1]^d$ for $i \rightarrow \infty$ in the $L^1$-Norm. Any ideas for such a sequence or how to ...
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Proof Clarification for Cantor Function is Holder Continuous

This is a follow up from the post: Cantor Function Clarification. For convenience I will copy the definition here again: Recall the Cantor set $C \subseteq [0, 1]$ is compact, has Hausdorff dimension ...
3 votes
1 answer
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Cantor Function Clarification

I am reading some lecture notes and came across the Cantor Function. However, I have some questions about it after reading. Recall the Cantor set $C \subseteq [0, 1]$ is compact, has Hausdorff ...
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2 answers
77 views

Observation on the ''explicit formula'' for the Cantor Set

Let us consider the closed interval $[a, b]$. Removing the open middle third interval in $C_0$, we let $C_1$ be the union of the remaining intervals, and have \begin{equation} C_1 = \left[a, a + \frac{...
2 votes
2 answers
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Counterexample: Continuous function with level sets of Lebesgue measure zero

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. It is given that it has "no flat regions", i.e. its level sets have empty interiors. My question is, must it mean that each of ...
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1 vote
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Countable dense subset of Cantor space isn't Wadge reducible to its complement

I am trying to prove that a countable dense subset $Q\subset 2^\omega$ (where $2^\omega$ denotes the Cantor space) is not Wadge reducible to its complement. I am trying to prove this because I want to ...
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2 answers
146 views

Is there a Cantor space that contains the interval?

Cantor spaces are topological spaces which are compact, totally disconnected, and perfect. The prototypical example is the Cantor set equipped with the Euclidean topology, although other such examples ...
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Explicit formula for the SVC Set

My goal is to arrive at an explicit formula for the $n$-th iteration of the algorithm that defines the Smith-Volterra-Cantor set (starting with $[0,1]$ and removing in the $n$-th step an open interval ...
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Box Dimension of Cantor Middle-Third Set

The calculation of the box dimension for the Cantor middle third set $C$ typically goes as such In the $n$th step of constructing $C$, there are $2^n$ line segments of length $\frac{1}{3^n}$ each ...
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6 votes
1 answer
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The Cantor set + the Cantor set = $[0,2]$

Let $C$ be the Cantor set. Prove $ C + C = [0,2]$. (Notation: If $S, R$ are sets, then $S + R$ is the set of all $s + r$ with $s \in S, r \in R$.) Proofs of this are readily available. This ...
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1 vote
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Covering zero-dimensional space with disjoint balls

Suppose that $(X, d)$ is compact zero-dimensional metric space. That it, $X$ has basis of clopen sets. Fix $\varepsilon > 0$. Can we find open cover consisting of disjoint balls of radius at most $\...
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Question on Hausdorff Measure and Dimension of the Cantor Set.

I’m trying to prove that the Hausdorff dimension of the triadic Cantor Set is $\frac{\ln(2)}{\ln(3)}$. The proof has been explained many times already on this site and relies on the fact that if $\...
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Initial Function in the Iterative Construction of the Cantor-Lebesgue Staircase Function

In the Wikipedia of the Cantor function, one iterative construction for it is presented there: Let $f_0(x)=x$ on $[0,1]$. For every positive integer $n$, define $$f_{n+1}(x)=\begin{cases} \frac{f_n(...
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What makes the proof of Cantor's theroem, for countably infinite sets, inapplicable to infinite sets?

(This may be a naive question born out of ignorance, so bear with me) What is the reason that Cantor's theorem can't be proven for infinite sets "simply" by indicating that for every element ...
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1 vote
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Openness/Closedness of the set of all ternary expansions with the same first n digits

Let $x \in [0,1]$ and consider the set of all ternary expansions in the interval $[0,1]$ and in the Cantor Set whose first n digits are equal to those of ternary expansion of $x$. I believe that for ...
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Proof of Continuity at endpoints for Cantor Lebesgue fn.

Here is the proof of continuity from the book of Royden "real Analysis" 4th edition: At the end of the proof, the author said that if $x_0$ is an endpoint then a similar argument ...
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Understanding how the Cantor set contains the point $4/9.$

Here is a statement from Bartle "Introduction to real analysis" that say that the Cantor set contains the point $4/9$ when $k=2$ and $n=2.$ I know from the first Cantor set $F_1,$ that we ...
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Understanding defn.of the Cantor- Lebesgue fn. $\varphi$ on the Cantor set $\textbf{C}.$

Here is the definition of $\varphi$ from Royden "Real Analysis" fourth edition: But I do not understand the definition of the Cantor- Lebesgue function $\varphi$ on the cantor set, can ...
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Prove the Cantor set doesn't contain any open interval [duplicate]

Need to prove that $C=\{ \sum _{k=1}^\infty \frac {a_k}{3^k} ; a_k \in \{0,2\}\}$ doesn't contain any open interval and I am not sure how to begin. Thank you
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Measure of Generalized Cantor Set plus Its Scaled Version

This is essentially Problem 10, Chapter 7 in Stein and Shakarchi's Real Analysis. Let $C_\xi$ be the generalized Cantor set obtained by repeatedly removing the centrally situatated open interval of ...
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Continuous mapping of the unit interval to a compact subset of $\mathbb{R}^d$

This is directly copied from Problem 6 in Chapter 7 of Stein and Shakarchi's Real Analysis. A compact subset $K$ of $\mathbb{R}^d$ is uniformly locally connected if given $\epsilon > 0$ there ...
5 votes
1 answer
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Is a subset of $\mathbb{R}$ that has the in-betweenness property, somewhere dense?

Suppose $ X \subset \mathbb{R},\ $ with $\ \vert X\vert \geq 3, $ has the property that: $$ \forall\ x,y \in X\ \text{ with} \ x<y,\ \exists\ z\in X\ \text{ such that } x<z<y.\qquad (\text{...
6 votes
2 answers
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Does Cantor's set contain a copy of each finite set?

Let $C$ denote the (usual) Cantor's set in the interval $[0,1]$. If $S=\{x_1,x_2,\cdots,x_n\}$ is a finite set of points in $\mathbb R$, is it true that $aS+b\subset C$ for some $a\neq 0$ and $b\in\...
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5 votes
1 answer
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No perfect non-empty set in $\mathscr{P}(\Bbb{R})$ can be strong measure zero.

Here is what to be proven: No perfect non-empty set in $\mathscr{P}(\Bbb{R})$ has strong measure zero. The textbook (Teoría de la Medida, Jaime San Martin Aristegui, section 1.6) suggests the ...
5 votes
1 answer
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Alternate forms of the Cantor Function? (aka Devil's Staircase)

$\def\R{\mathbf{R}}$ $\def\Q{\mathbf{Q}}$ Background: Using properties of uniform convergence and the definition of the function, in an exercise I proved that the Cantor Function is increasing, is ...
1 vote
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Existence of a function with a certain property on a subset of the cantor set

Let $\mathcal{C} \subset [0,1]$ denote the usual Cantor set (i.e. the usual definition based on removed middle third open intervals succesively). Define $S = \mathcal{C} - E$ where $E$ is the set $\{0,...
1 vote
1 answer
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Lebesgue's singular function nondecreasing

I'm reading Hewitt-Stromberg Real and Abstract Analysis. I fail Exercise (8.28). Here Lebesgue's singular function $\psi$ is first defined, after at $0$, on the open intervals excluded from Cantor's ...
2 votes
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What hypotheses do I need to add to prove that, The closure of a positive Lebesgue measure set contains an open set?

I know that the fat Cantor set is a counterexample to the above statement, but I would like to know if there is any hypothesis I could add to make this statement valid.
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Cantor Set : how did his base 3 system work?

I'm reading about Cantor Sets on Wikipedia. The article claims Cantor used a base 3 system for his theory, however I'm unable to find any material detailing how this system works. Can anyone point ...
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1 vote
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Apparent contradiction involving a.e. differentiability of strictly increasing functions

$U:[0,1]\rightarrow[0,1]$ is a continuous function, $U(1)>U(0)$ and $U'$ exists almost everywhere. Let $\max U([0,1])=\overline{v}$. Pick $\underline{v} \in (U(0), \overline{v})$. Define $\psi: [\...
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0 votes
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Cardinality of Cantor-like set for which deleted intervals have length $\frac{\alpha}{3}$ instead of $\frac{1}{3}$.

I have been reading the book Modern Real Analysis by Ziemer and have come to an exercise in the chapter on measure theory that I am having trouble with. The exercise concerns a Cantor-like set where ...
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Can we embed a countable compact subset of $\Bbb{Q}$ into the Cantor space? (Both use the standard topology.)

Let X be a compact subspace of $\Bbb{Q}$ with cardinality countable, denote the Cantor space as $\mathcal{C}$. Is there a subspace $Y \subset \mathcal{C}$ and a mapping $f$ such that $f: X \rightarrow ...
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0 votes
1 answer
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Perfect set with empty interior

Let $G\subset (-1,1)$ be a perfect, compact set with an empty interior. Since $(-1,1)\setminus G$ is an open set we can write it as follows $(-1,1)\setminus G=\cup (a_i,b_i)$, es decir $$G=(-1,1)\...
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(Generalized) Cantor set is uncountable

The standard Cantor set formed by recursively removing the middle one-thirds, on the interval $[0, 1]$ can be shown to be equal to the uncountable set of the base-3 numbers between $0$ and $1$ with ...
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1 vote
2 answers
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Is Cantor set is compact with respect to lower limit Topology on $R$?

Is Cantor set is compact with respect to lower limit Topology on $R$? I know Cantor set is compact with respect to usual topology. But I think it's not compact in lower limit Topology.. Because I ...
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Lebesgue-measure for closed subsets of [0,1]

I have to proof that: for each number $\alpha$ with $0<\alpha<1$ there is a closed subset $C$ of $[0,1]$ that satisfies $\lambda(C)=\alpha$ and includes no non-empty open set. I think that the ...
2 votes
2 answers
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Does the set of convex combination of points in Cantor set contains a non empty open interval?

$\mathcal{C}$ denote the cantor middle third set. $$\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$$ $\mathcal{C}_0=\mathcal{C}_1=\mathcal{C}$ and we can prove that that $\mathcal{C}$ contains no ...
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Give an example of a space that contains a unit interval and a dense set of non-isolated degenerate points

Let $X$ be a compact metric space. Let $X^{deg} = \{x \in X: \{x\}$ is a connected component$\}$. A space $X$ is said to be almost totally disconnected if the set $X^{deg}$ is dense in $X$. A point $x ...
1 vote
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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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2 votes
1 answer
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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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2 votes
1 answer
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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 ...
1 vote
1 answer
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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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4 votes
1 answer
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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 $...
1 vote
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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 ...
3 votes
1 answer
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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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1 vote
1 answer
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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 ...
-2 votes
1 answer
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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{...
0 votes
1 answer
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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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