Cantor Set - Perfect and Totally Disconnected

  • #1
I am getting ready for grad school in the fall, and re-teaching myself a bunch of undergraduate subjects. Right now I am reading up on the topology of the real number line. I have come across a fact that is really difficult for me to wrap my head around:

The Cantor set is both perfect, and totally disconnected.

I am aware that this is the only set (up to homeomorphism) that has both of these properties simultaneously. I guess my question is:

Can someone explain to me how it is possible for a set to be both Perfect and totally-disconnected?

I am not looking for a proof. I read and understand the proofs, I just can't believe what the logic is telling me. Instead, I am looking for more of a informal explanation. How can a set be closed with no isolation points, but at the same time be totally disconnected!?
 

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  • #2
lavinia
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[ How can a set be closed with no isolation points, but at the same time be totally disconnected!?

Not sure what intuition you are looking for.

The Cantor set is the complement of an open set and so it closed. Since it has measure zero it can not contain any connected subsets other than points since all of the connected subsets of the line are intervals.I guess the hard part is visualizing why every point in it is a limit point.

What about the Cantor sets of positive measure? For instance, instead of removing successive middle thirds, remove middle fifths.
 
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  • #3
it can not contain any connected subsets other than points

Wouldn't this mean that the points are isolated?

I guess the hard part is visualizing why every point in it is a limit point.

This is exactly what I mean! The problem seems to be if you assume it is totally disconnected, and you assume it's closed, it's hard to imagine it has no isolation points. (Which is the point you were making)

On the other hand, if you assume it is totally disconnected, and then assume each point is a limit point, then it is hard to imagine that it is closed. Given two points in the Cantor set, wouldn't there need to be some point in between that is not in the Cantor set in order to write it as the union of two disconnected sets!? But then, if each point is limit point, then there are limit points that are not contained the set, so it can't be closed!

Take the rational numbers for example. It is easy enough to see they are totally disconnected, and each point is a limit point, but the set is certainly not closed, since each irrational number is a limit point as well.
 
  • #4
lavinia
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Wouldn't this mean that the points are isolated?

No for instance the sequence n-1/n together with the number, 1 is totally disconnected but 1 is not isolated.
 
  • #5
No for instance the sequence n-1/n together with the number, 1 is totally disconnected but 1 is not isolated.

But each of the numbers here is not a limit point, so there is always an irrational number you can use as the endpoints of the separated sets.

I appreciate the explanations, I guess I am still confused though. Hopefully, I can wrap my head around this once I become more familiar with other properties of the real numbers. The Cantor set is definitely a very interesting and counter-intuitive subset of the real numbers.
 
  • #6
lavinia
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But each of the numbers here is not a limit point, so there is always an irrational number you can use as the endpoints of the separated sets.

I appreciate the explanations, I guess I am still confused though. Hopefully, I can wrap my head around this once I become more familiar with other properties of the real numbers. The Cantor set is definitely a very interesting and counter-intuitive subset of the real numbers.

All I was pointing out is that the number 1 is no isolated from the other points in the sequence. Any open interval around it contains points in the sequence. It does not matter how narrow the interval is. The other points are isolated.

But the example of the number 1 in this sequence is exactly what happens for every point in the Cantor set. each point in the Cantor set is like the number 1 in this example.
 
  • #7
But the example of the number 1 in this sequence is exactly what happens for every point in the Cantor set. each point in the Cantor set is like the number 1 in this example.

OK, that makes it a little more accessible. Thanks!
 
  • #8
lavinia
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here are some examples of points in the Cantor set that are not isolated.

Write the Cantor set in base 3. Any trecimal consisting completely of zeros and twos is in the Cantor set. Further any trecimal with a 1 but all zeros afterwards is in the Cantor set.

Examples .1000000000 This is the left end point of the first middle third

Any interval around this point contains infinitely many points in the sequence .02 .022

... .0222222222222 n times. All of the points in the sequence lie in the Cantor set.

- take any point that has a trecimal expansion with infinitely many 2's in it. Then any neighborhood of it contains infinitely many points in the sequence that takes the first n digits of the number and is all zero afterwards.
 
  • #9
That's a great way to visualize the Cantor set. I would have never thought of writing in in base 3 - but it makes so much sense to do so once you pointed it out
 

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