# Homework Help: Cantor sets (poof needed)

1. Oct 11, 2011

### sazanda

1. The problem statement, all variables and given/known data

Let C be a Cantor set and let x in C be given
prove that
a) Every neighborhood of x contains points in C, different from x.
b) Every neighborhood of x contains points not in C

2. Relevant equations

How can I start to prove?

3. The attempt at a solution

n/a

2. Oct 11, 2011

### Dick

The usual way would be to think about what the Cantor set means in terms of an expansion in base 3, i.e. ternary.

3. Oct 12, 2011

### sazanda

I need some kind of initiation. Prof. had just defined the Cantor set and assigned this problem.

4. Oct 12, 2011

### HallsofIvy

Okay, what definition did your teacher give you? And are you talking about a Cantor set or the Cantor set?

5. Oct 12, 2011

### sazanda

a cantor set

6. Oct 12, 2011

### HallsofIvy

Okay, but once again, what definition of "Cantor set" are you using?

7. Oct 12, 2011

### sazanda

A1={[0,1/3],[2/3,1]}
A2={[0,1/9],[2/9,3/9],[6/9,7/9],[8/9,9/9]}
:
:
:

intersection of all Ai is the Cantor set.

This is definition that Prof. defined.

8. Oct 12, 2011

### Dick

The hint in post 2 remains valid. You might also think about what the measure (length) of the Cantor set might be.

9. Oct 12, 2011

### HallsofIvy

Do you see that, in base 3 notation, every number in [0, 1/3] starts "0.0..." and every number in [2/3, 1] starts "0.2..". That is, A1 contains all real numbers between 0 and 1 whose base 3 representation does NOT have a "1" in the first place.

Do you see that, again in base 3 notation, every number in [0, 1/9] starts 0.00..., every number in [2/9, 3/9] starts 0.02..., every number in [6/9, 7/9] starts 0.20..., and every number in [8/9, 1] starts 0.22... That is, A2 contains all real numbers between 0 and 1 whose base 3 representation does NOT have a "1" in the first two places.

Ai is the set of all numbers between 0 and 1 whose base three representation does not have a "1" in the first i places.

So the Cantor set is the set of all numbers between 0 and 1 whose base 3 representation does not contain any "1" in its base three representation. Now look at intervals around a number "x" in that set.

10. Oct 12, 2011

### Dick

You could also think about endpoints. Did you notice all of the endpoints of the intervals in A_n are also in A_n+1?