Flashcards: The Cantor Set — 28 cards

What is the Cantor set in [0, 1]?

A subset obtained by repeatedly removing the open middle third from every remaining closed interval.

Who introduced the Cantor set and in which year?

Georg Cantor, 1883.

What is the initial interval in the Cantor set construction?

The interval $I_0=[0,1]$.

Which open interval is removed first in the Cantor set construction?

The open interval $(1/3,2/3)$ is removed first.

How is the Cantor set defined using the intervals $I_n$?

As the intersection $C=\bigcap_{n=1}^{\infty} I_n$.

Why is the Cantor set non-empty?

Because every endpoint of every closed interval at any stage belongs to it.

Why is the Cantor set a closed subset of [0, 1]?

Because it is the intersection of finite unions of closed intervals.

How many closed intervals does set $I_n$ have at stage $n$?

It has $2^n$ closed intervals.

What is the length of each interval in set $I_n$ at stage $n$?

Each interval has length $1/3^n$.

What is the formula for the length $\ell(I_n)$ of the stage-$n$ set?

$\ell(I_n)=2^n \cdot (1/3^n) = (2/3)^n$.

Why does the Cantor set have length zero?

Because $\ell(C)=\lim_{n\to\infty}(2/3)^n=0$.

What does every point of the Cantor set determine?

A unique infinite sequence of left and right choices through nested intervals.

How is the left/right sequence converted into a ternary expansion?

By replacing left with 0 and right with 2.

How is the sequence $b=b_1b_2b_3\ldots$ constructed from $a_1,a_2,a_3,\ldots$ in the uncountability proof?

Choose $b_i=0$ if $a_{ii}=2$ and $b_i=2$ if $a_{ii}=0$.

What does the diagonal construction produce in the Cantor set proof?

An element differing from the nth listed element at its nth ternary digit.

What does the diagonal construction prove about the Cantor set?

That the Cantor set is uncountable.

What does the Cantor set contain regarding intervals?

It contains no interval.

What shows a real number can have multiple ternary expansions?

The equality $(0.1)_3=(0.022\ldots)_3=1/3$ shows this.

When does a number belong to the Cantor set in ternary terms?

If its ternary expansion contains only digits 0 and 2.

How does induction prove a number with ternary digits 0 and 2 belongs to the Cantor set?

It shows the number belongs to every stage $I_k$ and thus to $C$.

What are the intervals after the first stage in Cantor set construction?

$I_1=[0,1/3]\cup[2/3,1]$.

What are the intervals after the second stage in Cantor set construction?

$I_2=[0,1/9]\cup[2/9,3/9]\cup[6/9,7/9]\cup[8/9,9/9]$.

How many points does the Cantor set contain?

Countably many points.

Which points of the form $1/3^n$ belong to the Cantor set?

Points for all $n=1,2,3,\ldots$ belong to it.

What does it mean for the Cantor set to be nowhere dense in [0, 1]?

It is a subset of [0, 1] with empty interior.

What does totally disconnected mean for the Cantor set?

It has no connected subset with more than one point.

What is the compactness status of the Cantor set in [0, 1]?

It is compact in the usual topology on [0, 1].

What is the formula for a number with ternary digits $a_k \in \{0,2\}$?

$x=\sum_{k=1}^{\infty}a_k3^{-k}$