# 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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 ...