# 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.

359 questions
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### To prove that Cantor function is not absolute continuous, do we need to take countable infinite number of subintervals?

This highly upvoted answer claims that we can show that the Cantor function is not absolute continuous by taking a finite number of subintervals. However, one comment claims that the "answer" is ...
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### How to calculate the weight of the Cantor dust?

(or "What is wrong in cardinality × size of the elements of the Cantor set?") Supposing the "standard" Cantor set definition, $$\mathcal{C} = \bigcap_{k=1}^\infty C_k$$ where $|C_k|=2^k$, the ...
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### Why Cantor set is not definied by a limit?

The Cantor set, as showed in books and Wikipedia, is defined in terms of $C_k$, the finite Cantor set of level $k$: $$\mathcal{C} = \bigcap_{k=1}^\infty C_k$$ But after intersection only the "last"...
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### Using union of countably infinite sets, I tried to prove that set of all real numbers in [0,1) is countable

Cantor's diagonal method shows that the set $S=\{x\in \Bbb R|x \in [0,1)\}$ is uncountably infinite, because there is no bijection between the set $S$ and the set of natural number $\Bbb N$. I came ...
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### prove that the ternary cantor set is compact and a perfect set.

prove that the ternary cantor set is compact and a perfect set. My trial: I know that I should prove that it is closed and bounded, for proving that it is closed because finite union of closed sets (...
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### Cardinality of Sub(A) is smaller than ${\aleph_0}$ or equal $2^{\aleph_0}$?

Let $\mathbb{A}=(A,\mathcal{F}^\mathbb{A})$ be a countable algebra. I need to prove that $|Sub(\mathbb{A})|\leqslant\aleph_0$ or $|Sub(\mathbb{A})|=2^{\aleph_0},$ where $Sub(\mathbb{A})$ is the ...
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### Explicit homeomorphism between product of Cantor sets onto the Cantor set

I want to find an explicit homeomorphism $\varphi: C\times C \longrightarrow C$ where $C$ denotes the Cantor set. The hint is to use the base $3$ expansion of the elements of the Cantor set. My two ...
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### Let $X$ be the topological space formed by $K$ and an isolated point $p \notin K$. Show that $X \cong X^2$ but $X \ncong X^n$ [closed]

K is the usual Cantor set. I was studying and I came across this problem, but I could not solve it.
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### Homeomorphism from the real numbers to the real numbers with restriction to the Cantor set.

Let $K$ be the Cantor set and $C \subset R$ be a non empty compact set with no isolated points and empty interior. Prove that it exists a homeomorphism $f:R \longrightarrow R$ such that $f(K)= C$.
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### Convolution square of the Cantor set

For $0\leq d\leq 1$, let $\eta_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$ (for some normalization); recall that it is translation-invariant. Motivation for what follows: Up to ...
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### Quotient space $[0,1]/C$ is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,…\right\}$, $C$ denotes Cantor set. [closed]

How to prove quotient space $[0,1]/C$, where $C$ denotes Cantor set, is homeomorphic to $[0,1]/\left\{0,1,\frac12,\frac13,...\right\}$?
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### G is the Cantor set, $X$ is a countable set of G hence G\X is dense

Exercise: Let $G$ be a Cantor set. a) Prove that $G$ has a countable dense subset. b)If $X$ is a countable subset of $G$ then $G\setminus X$ is dense in $G$ a) It is know that the Cantor ...
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### The cantor set $K$ contains no intervals

Let $K$ the Cantor set. I already proved the following properties Properties.\begin{equation} \begin{split} (1)&\quad|K|=|\mathbb{R}|\\ (2)&\quad\lambda(K)=0,\text{where}\;\lambda\;\text{is ...
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### Neutral element for a Cantor Set.

On Wiki I found the following statement: $T_L$ and $T_R$ together with function composition forms a monoid. I am able to prove the associativity of the composition operation, but what will be a ...
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### Koch curve from Cantor sets (paradox)

The Koch curve is normally constructed by taking a line segment, replacing the middle third with two copies of itself forming legs in an equilateral triangle, and then repeating this recursively for ...
### Prove that the Cantor set is homeomorphic to $(X,\mathscr T)$.
For each $n\in \mathbb N$, let $X_n=\{0,2\}$ and let $\mathscr T_n$ be the discrete topology on $X_n$. Let $X=\prod_{n=1}^\infty X_n,$ and $\mathscr T$ be the product topology on $X$. Prove that the ...