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

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    Cantor Set
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Discussion Overview

The discussion centers on the properties of the Cantor Set, particularly regarding the nature of its interior points and endpoints. Participants explore the implications of the Cantor Set's construction, its measure, and the classification of points within it.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the Cantor Set being a collection of closed sets and its measure being zero, leading to questions about the nature of endpoints and interior points.
  • Another participant asserts that the Cantor Set does not contain any intervals, challenging the notion of endpoints associated with points in the set.
  • A different participant questions the validity of defining an interval of measure zero, suggesting that intervals must have non-zero measure based on their definition.
  • There is uncertainty expressed about whether 1/4 is an interior point of the Cantor Set, with one participant stating that any neighborhood of 1/4 contains points not in the Cantor Set.
  • One participant references the Wikipedia page on the Cantor Set and expresses doubt about its accuracy, mentioning a distinction between frontier points and other types of points in the set.
  • Another participant provides the base 3 expansion of 1/4 as a means to determine its membership in the Cantor Set.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the classification of points in the Cantor Set, particularly regarding the status of 1/4 as an interior point and the existence of endpoints. Multiple competing views remain on these topics.

Contextual Notes

There are limitations in the discussion regarding the definitions of endpoints, boundary points, and frontier points, as well as the implications of measure zero in the context of intervals.

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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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.
 
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.
 
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
 
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.
 
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
 
well, I think I get it, thanks a lot
 

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