Exploring the Cantor Set: Why There Are No Interior Points

In summary: This means that in the Cantor set, there are no finite trinary numbers but there can be infinite trinary numbers such as 0.1 or 0.222222... In summary, the Cantor set is a set of real numbers that is constructed by removing the middle third of intervals infinitely, resulting in a set with no interior points. This is because every point's neighborhood becomes incomplete due to the removal process. The measure of the Cantor set is zero, and in the trinary system, the terms in the Cantor set have no 1's in their base-3 expansion. However, there can be infinite trinary numbers in the set, such as 0.1 or 0.222222..., which explains
  • #1
bedi
81
0
Why there is no interior points in a Cantor set? Please explain me in detail.
 
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  • #2
What did you try already?
 
  • #3
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?
 
  • #4
What is the measure of the Cantor set?
 
  • #5
I don't know that measure thing yet...
 
  • #6
bedi said:
I don't know that measure thing yet...

OK. Tr describing the Canor set in the trinary system.
 
  • #7
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.
 
  • #8
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...
 
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What is the Cantor Set?

The Cantor Set is a set of numbers created by removing the middle third of a line segment repeatedly, resulting in an infinite set of points that are not connected. It is named after German mathematician Georg Cantor.

Why are there no interior points in the Cantor Set?

There are no interior points in the Cantor Set because every point in the set is an endpoint of a line segment that has been removed during the construction process. This means that there are no points that are completely contained within the set.

How is the Cantor Set related to the concept of self-similarity?

The Cantor Set is an example of a self-similar set, which means that it has the same shape at different scales. This is because the process of removing the middle third of a line segment is repeated infinitely, resulting in smaller and smaller line segments that have the same shape as the original set.

What is the dimension of the Cantor Set?

The Cantor Set has a dimension of 0, which is known as a zero-dimensional set. This is because it has no interior points and cannot be measured using traditional methods of dimensionality.

What are some real-world applications of the Cantor Set?

The Cantor Set has been used in various fields, including computer science, physics, and biology. In computer graphics, it can be used to create fractal patterns, and in physics, it has been used to model diffusion and phase transitions. In biology, it has been used to study the hierarchical structure of protein molecules.

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