Is the Completeness Axiom Equivalent to the Heine Borel Theory?

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Discussion Overview

The discussion revolves around the relationship between the Completeness Axiom and the Heine-Borel Theorem, specifically whether the Completeness Axiom can be used to prove the Heine-Borel Theorem regarding open finite covers. Participants explore the implications of the Completeness Axiom in relation to closed and bounded sets.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to prove the equivalence of the Completeness Axiom and the Heine-Borel Theorem, questioning whether it is necessary to assume that the set is closed when transitioning from the Completeness Axiom to Heine-Borel.
  • Another participant clarifies that the Completeness Axiom refers to the existence of upper and lower bounds for sets of real numbers, and questions the assumption of closed sets in this context.
  • A participant asserts that the Heine-Borel Theorem states the equivalence between compactness and closed, bounded subsets of R^n, and does not inherently assume a set is closed.
  • There is a suggestion that in proving the Heine-Borel Theorem, one must assume that the set is closed and bounded, while also utilizing the Completeness Axiom in the proof.
  • Confusion arises regarding the necessity of assuming a set is closed, with participants expressing uncertainty about the implications of the Completeness Axiom on this assumption.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of assuming that sets are closed when discussing the Completeness Axiom and its relation to the Heine-Borel Theorem. The discussion remains unresolved regarding the implications of these assumptions.

Contextual Notes

Participants highlight the importance of definitions and assumptions in their arguments, particularly regarding the nature of sets being closed and bounded in relation to the Completeness Axiom and the Heine-Borel Theorem.

autobot.d
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Hello,

I am trying to prove that The Completeness Axiom is equivalent to the Heine Borel Theory (open finite cover).

I was wondering when going from Completeness Axiom to Heine Borel, is ok to assume that the set is closed?
I know the heine borel assumes that it is closed, but Completeness does not. I was just wondering if I had to prove that Completeness also asserted that the set was closed.

Thanks for the help.
 
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What is your "completeness axiom"? That every set of real numbers bounded from above (below) has an upper (lower) bound? What set do you mean when you say "that Completeness also asserted THE set was closed"?

Also, Heine-Borel states equivalence between compact and (closed, bounded) subsets of R^n. It does not "assume" some set is closed.
 
That be the completeness axiom I am talking about.

Not sure what you mean by not assuming set is closed.

"Let F be a closed and bounded set of real numbers. Then every open cover of F has a finite subcover."

Thanks for the help!
 
Let me put it this way: yes, you have to assume F is closed and bounded, and then in your proof use the completeness axiom somewhere. The completeness axiom does not say anything about a set being closed.
 
Ahhh.

So I can only use the fact that my set is bounded, if I have you correct.
That is what I figured. Thank you again for your help.
 
landau said:
you have to assume f is closed and bounded
autobot.d said:
so i can only use the fact that my set is bounded, if i have you correct.
I don't see how you get that from my answer.
 
The completeness axiom does not say anything about a set being closed.

I thought you meant that proving the Completeness Axiom using Heine Borel Theorem that I cannot use that the set is closed since the Completeness Axiom doesn't call for it. Sorry for the confusion.
 

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