Clarifications regarding the reals

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Discussion Overview

The discussion revolves around the concept of sequential compactness in the context of the real numbers, specifically addressing the definitions and implications of sequential compactness and compactness in topological spaces. The scope includes theoretical clarifications and definitions related to mathematical concepts.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states that since \(\mathbb{R}\) is complete, every Cauchy sequence converges, leading to the assertion that every subsequence also converges.
  • The same participant questions the definition of sequential compactness, suggesting that it seems disconnected from the concept of compactness until a later theorem is introduced.
  • Another participant counters that the sequence \(1, 2, 3, 4, \ldots\) in \(\mathbb{R}\) does not have a convergent subsequence, challenging the initial claim.
  • A third participant clarifies that \(\mathbb{R}\) is not sequentially compact because not every sequence has a convergent subsequence, noting that only bounded sequences do.
  • This participant also corrects the definition of sequential compactness, emphasizing that it applies to closed and bounded sets, which aligns with the compactness theorem.
  • A later reply expresses gratitude for the clarifications, indicating a recognition of misinterpretation of the definitions.

Areas of Agreement / Disagreement

Participants express disagreement regarding the sequential compactness of \(\mathbb{R}\), with some asserting it is not sequentially compact due to the lack of convergent subsequences in certain sequences, while others initially suggested a misunderstanding of the definitions. The discussion remains unresolved regarding the implications of the definitions.

Contextual Notes

The discussion highlights limitations in the understanding of definitions related to sequential compactness and compactness, particularly in the context of boundedness and closure of sets.

Bachelier
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We know [tex]\mathbb{R} \ or \ \mathbb{R}^1[/tex] is complete, hence every Cauchy seq. converges. Therefore every subsequence of them converges as well. (to the same pt)

My clarification is regarding the term Sequentially Compact (SC). In the definition, they only state a topological space X is SC if every sequence in X has a convergent subsequence.

I know there is the term compact in the name but it is never mentioned in the definition. Only later do we make the connection between compactness and SC via the theorem: X is SC iff it is compact. Which begs the question that the definition of sequentially compact has nothing to do with compactness in the first place.

So if it doesn't and the relationship between the two properties is only proven later after introducing both terms separately, then based on these definitions, [tex]\mathbb{R}[/tex] is sequentially compact. And therefore it is compact per theorem, which we know is not true because it is unbounded. :confused:
 
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The sequence 1,2,3,4,... in R had no convergent subsequence.
 
No, R is NOT "sequentially compact" for two reasons.

1) It is not true that "every sequence of real numbers has a convergent subsequence". For example, the sequence {1, 2, 3, 4, ...} of all positive integers has no convergent subsequence. What is true is that every bounded sequence of real numbers has a convergent subsequence.

2) You have stated the definition of "Sequentially Compact" slightly incorrectly: a set is "sequentially compact" if and only if every sequence of points in that set converges to a point in the set. That will be the same as saying "every sequence has a convergent subsequence" if the set is closed.

That is, every set of real numbers will be "sequentially compact" if and only if it is both closed and bounded, which, of course, also implies "compact".
 
Great. Thank you guys. I knew I was misreading the definitions.
 

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