How can I prove the properties of points in a Cantor set?

[sazanda]

Homework Statement

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

Homework Equations

How can I start to prove?

The Attempt at a Solution

n/a

 

[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.

 

[sazanda]

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

I need some kind of initiation. Prof. had just defined the Cantor set and assigned this problem.
I do not have more info about this. I looked the internet they are little bit complicated.

 

[HallsofIvy]

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

 

[sazanda]

a cantor set

 

[HallsofIvy]

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

 

[sazanda]

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

A₁={[0,1/3],[2/3,1]}
A₂={[0,1/9],[2/9,3/9],[6/9,7/9],[8/9,9/9]}
:
:
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intersection of all A_i is the Cantor set.

This is definition that Prof. defined.

 

[Dick]

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 A_i is the Cantor set.

This is definition that Prof. defined.

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

 

[HallsofIvy]

A₁={[0,1/3],[2/3,1]}

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.

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

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.

:
:
:

intersection of all A_i is the Cantor set.

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

This is definition that Prof. defined.

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.

 

[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?