What are the endpoints of the interior points in the Cantor Set?

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In summary, the Cantor Set is a collection of closed sets. It has measure zero and contains only intervals of zero measure. It also has interior points like 1/4, which are never endpoint. The endpoints associated with 1/4 may be in the neighborhood of 1/4, but it must be closed in order to be considered an open ball.
  • #1
Bob3141592
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The Cantor Set is making me very confused. I can understand that since only open sets are removed, the Cantor Set if a collection of closed sets. I believe I understand that the Cantor Set has measure zero, and therefore contains only intervals of zero measure. I can see that the endpoints of the segments left behind are never removed, and that there are a (countably) infinite number of them. What I didn't realize until recently is that not every point in the Cantor Set is an endpoint, but it also contains interior points, like 1/4, which is never an endpoint. That makes the Cantor Set uncountable infinite. What confuses me are the endpoints associated with that interior point. Since all intervals in the set must have 0 measure, I think there cannot be an explicit point other than 1/4 to be the boundary of the interval. But if the boundary point cannot be definitively named, in what sense can we call that interval closed? This is where I get all confused. I can only imagine the endpoints associated with 1/4 to be in the neighborhood of 1/4, and so it seems like the definition of an open ball about that point. But it has to be closed, and I can't get a handle on this.

Any help in seeing how this works would be greatly appreciated.
 
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  • #2
The cantor set doesn't contain any intervals (except for its individual points), and there's no "endpoints" associated with arbitrary points in the set.
 
  • #3
well, I'm not sure if it is okay to find a interval of measure zero, since interval is defined(imo) as [a,b] where a<b, a,b is in R, and therefore its measure is not zero.
What I am sure is that 1/4 is not an interior point since any neighborhood of it contains points not in Cantor Set.
 
  • #4
oh, could anyone remind me how to determine if 1/4 is in Cantor Set or not?...I got stuck on this...
Thanks a lot
 
  • #5
Perhaps the Wikipedia page on the Cantor Set is wrong. That was the basis of my information, and it didn't quite seem right to me, but I presumed it was my understanding that was wrong and not the page itself.

The Mathematics dictionary by James and James says all points in the Cantor Set are "frontier" points but I'm hazy on the distinctions between endpoints, boundary points and frontier points, if any.
 
  • #6
boombaby said:
oh, could anyone remind me how to determine if 1/4 is in Cantor Set or not?...I got stuck on this...
Thanks a lot

the base 3 expansion of 1/4 is 0.02020202..., which doesn't contain any 1s
 
  • #7
well, I think I get it, thanks a lot
 

1. What is the Cantor Set?

The Cantor Set is a mathematical construct created by the German mathematician Georg Cantor in the late 19th century. It is a set of points on a line that is constructed by repeatedly removing the middle third of each line segment.

2. How is the Cantor Set related to fractals?

The Cantor Set is considered to be one of the earliest examples of a fractal, which is a geometric shape that displays self-similarity at different scales. The Cantor Set has a similar pattern of self-similarity at different levels of magnification, making it a fractal.

3. What is the cardinality of the Cantor Set?

The Cantor Set has the same cardinality (or size) as the real numbers between 0 and 1, which is known as the cardinality of the continuum. This means that even though the Cantor Set has an infinite number of points, it is still considered to be a "smaller" infinity compared to the set of all real numbers.

4. How is the Cantor Set used in mathematics?

The Cantor Set has applications in various branches of mathematics, such as topology, measure theory, and dynamical systems. It is also used as an example of a set with interesting properties, such as being uncountable and having a non-zero measure despite being made up of points with no length.

5. Can the Cantor Set be generalized to higher dimensions?

Yes, the Cantor Set can be extended to higher dimensions to create the Cantor Cube or the Sierpinski carpet. These constructions follow a similar pattern of repeatedly removing smaller and smaller portions of a shape to create a fractal set with interesting properties.

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