Why Are There No Interior Points in the Cantor Set?

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Discussion Overview

The discussion revolves around the question of why there are no interior points in the Cantor set, exploring concepts related to measure theory, base-3 representation, and the properties of the set itself. Participants seek clarification on the nature of the Cantor set and its implications in a mathematical context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asks for a detailed explanation of the absence of interior points in the Cantor set.
  • Another participant questions what prior attempts have been made to understand the topic.
  • A participant reflects on their understanding, suggesting that the removal of the middle third of intervals leads to neighborhoods that are "incomplete," but expresses uncertainty about this reasoning.
  • There is a question about the measure of the Cantor set, indicating a potential interest in its mathematical properties.
  • Some participants indicate they are not familiar with the concept of measure, suggesting a gap in understanding that may affect the discussion.
  • A participant introduces the idea of describing the Cantor set in the trinary system, noting that points in the Cantor set have no '1's in their base-3 expansion.
  • A challenge is posed to show that for any point in the Cantor set, there exists a nearby point whose decimal expansion contains a '1', hinting at the density of numbers outside the Cantor set.
  • A participant raises a question about the inclusion of 1/3 in the Cantor set, discussing its representation in the trinary system and reflecting on the nature of finite trinary numbers.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the Cantor set and its properties. There is no consensus on the explanations provided, and some participants remain uncertain about key concepts, indicating that multiple views and interpretations are present.

Contextual Notes

Some participants express a lack of familiarity with measure theory, which may limit their understanding of the discussion. The exploration of the Cantor set in the trinary system introduces additional complexity that is not fully resolved.

bedi
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Why there is no interior points in a Cantor set? Please explain me in detail.
 
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What did you try already?
 
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?
 
What is the measure of the Cantor set?
 
I don't know that measure thing yet...
 
bedi said:
I don't know that measure thing yet...

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