# 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 ...
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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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### 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{...
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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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### 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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### 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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### 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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### 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 ...
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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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### 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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