Articles of cantor set

the elements of Cantor's discontinuum

Let $(A_n)_{n \in \mathbb{N}}$ the sequence of subsets of $\mathbb{R}$, given by $A_0 := \bigcup_{k \in \mathbb{Z}}[2k, 2k + 1]$ und $A_n := \frac{1}{3}A_{n-1}$ for $n ≥ 1$. Also, we define $$ A := \bigcap_{n=0}^\infty A_n, \,C:= A \cap [0, 1]$$ We call $C$ Cantor’s discontinuum. Given this definition, I now want to prove that […]

Cantor set: Lebesgue measure and uncountability

I have to prove two things. First is that the Cantor set has a lebesgue measure of 0. If we regard the supersets $C_n$, where $C_0 = [0,1]$, $C_1 = [0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and so on. Each containig interals of length $3^{-n}$ and by construction there are $2^n$ such intervals. The lebesgue measure of each […]

Continuous functions and uncountable intersections with the x-axis

Let $f : \mathbb{R} \to \mathbb{R}$ such that the set $X = \{x \in \mathbb{R} : f(x) = 0\}$ does not contain any interval (i.e. there is no interval $I \subset X$) Of course the set $X$ can be uncountable (see Cantor Set). If we add that $f$ is continuous, is it true that X […]

Does every continuous map between compact metrizable spaces lift to the Cantor set?

I’m interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is “universal” in the category of metrizable compact spaces, in the sense that every compact metrizable space is a quotient of $2^\mathbb{N}$ (although the quotient map is not unique). My main question is: Does every continuous map […]

Find a Continuous Function with Cantor Set Level Sets

This was a problem from a class that I thought was really interesting. It asked to find function $f\in C[0,1]$ such that the sets $\{x:f(x)=c\}$ form a Cantor Set for all $0\leq c\leq 1$. I found a non-constructive proof of the existence of such functions, but would be curious if anyone could give a constructive […]

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a homeomorphism of the Cantor set? More precisely, let $K$ be a compact metric space, and let $h\colon K\to […]

Pairing function for ordered pairs

Is there a pairing function like Cantor’s ( that would map ordered pairs (of integers) to different integers? ie: (M, N) -> L1 (N, M) -> L2 Where L1 != L2 All input integers could be positive, but the output does not have to be perhaps something like: newpair(M, N) = if (M < […]

Dense subset of Cantor set homeomorphic to the Baire space

Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.

Is the Cantor set made of interval endpoints?

The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably many intervals, and conclude the Cantor set is countable?

The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself – cylinder basis – and it topology

I know the Cantor set probably comes up in homework questions all the time but I didn’t find many clues – that I understood at least. I am, for a homework problem, supposed to show that the Cantor set is homeomorphic to the infinite product (I am assuming countably infinite?) of $\{0,1\}$ with itself. So […]