Is the Standard Proof of [a,b] in R Being Compact Flawed?

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Discussion Overview

The discussion revolves around the standard proof that the closed interval [a,b] in R is compact within the context of topology. Participants are examining the validity of the proof, particularly the method of dividing the interval and the implications of uncoverable halves at each step.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the proof's assumption that at least one half of the interval remains coverable by a finite subcollection of any open cover, suggesting that both halves could be uncoverable.
  • Another participant proposes that if both halves are uncoverable, one can arbitrarily choose either half for further analysis.
  • A third participant challenges the need for "picking" between halves if both are uncoverable, implying a potential flaw in the reasoning.
  • A later reply speculates that the proof may involve constructing a nested sequence of uncoverable intervals, emphasizing the importance of clearly stating each step in the argument.

Areas of Agreement / Disagreement

Participants express differing views on the implications of uncoverable intervals and the necessity of choosing between them, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants do not clarify the specific proof being referenced, and there is ambiguity regarding the assumptions made about uncoverable intervals and the steps involved in the proof.

ice109
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I have a bone to pick with the standard proof of the closed interval in R being compact with respect to the usual topology.

The proof starts out claiming that we can divide the interval in two and it is one of these two halves that is not coverable a finite subcollection of any open cover. Then we proceed to cut that interval in half and make the same claim and so on and so on.

The bone that I pick is what if both halves of the initial interval are uncoverable or both quarters of the initial half and so on.
 
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If both halves are uncoverable at any step, pick either one.
 
then why even have any discussion about "picking."?
 
I don't know exactly what proof you are using but I'm guessing that you are showing you can construct a nested sequence of uncoverable intervals whose lengths go to zero, probably looking for a contradiction in an indirect argument. So you just have to say how you do each step. You always pick the "uncoverable" half or, if both halves are uncoverable, pick either one because it doesn't matter.
 

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