Why Are There No Interior Points in the Cantor Set?

[bedi]

Why there is no interior points in a Cantor set? Please explain me in detail.

 

[micromass]

What did you try already?

 

[bedi]

I've been watching a lecture and didn't understand why there wouldn't be any point whose neighbourhood is completely surrounded by the cantor set. Oh I think I start to grasp it now, since every interval's "middle" is removed and that process goes on forever, every point's neighbourhood becomes somewhat "incomplete". Am I wrong?

 

[lavinia]

What is the measure of the Cantor set?

 

[bedi]

I don't know that measure thing yet...

 

[lavinia]

I don't know that measure thing yet...

OK. Tr describing the Canor set in the trinary system.

 

[Bacle2]

Notice that like Lavinia said,the terms in the (Standard) Cantor set C have no 1's in

their base-3 expansion. Now try to show,given x in C --so that there are no

1's in the decimal expansion of x -- that, no matter how close you go about x in

(x-e,x +e ) , you will hit a number y in (x-e,x+e) ,whose decimal expansion _does_

have a 1 in it . Hint: you can cut-off the decimal expansion of x at any point,

as far back as you want.

 

[bedi]

Thank to all of you. But 1/3 is included in the cantor set and in trinary it's 0.1 isn't it? I see there is no finite trinary number in the set but why is 0.1 included?

edit: Now I think I'm okay. In the trinary number system, 1/3 can be written not only as 0.1 but also as 0.222222...