Math Problem Statement
If every closed and bounded subset of a metric space M is compact, does it follow that M is complete? (Proof or counterexample.)
Solution
Yes, it does follow that is complete if every closed and bounded subset of the metric space is compact. Here's a proof by contrapositive:
Contrapositive Statement:
We will prove the contrapositive of the original statement: If is not complete, then there exists a closed and bounded subset of that is not compact.
Proof:
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Assume is not complete. This means there exists a Cauchy sequence in that does not converge to a point in .
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Let be a Cauchy sequence in that does not converge in . Since is a Cauchy sequence, it is bounded (Cauchy sequences are always bounded in metric spaces).
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Now, consider the set . This set is closed because any limit point (if it exists) of the sequence must belong to , and all points are in . However, since the sequence does not converge in , no limit point actually exists in , and hence does not have any accumulation point that lies in .
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Thus, the set is closed and bounded but not compact. In a metric space, compactness requires that every sequence has a convergent subsequence, but since the sequence does not converge, there is no subsequence that converges to a point in .
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Therefore, is a closed and bounded subset of that is not compact.
This proves the contrapositive: If is not complete, there exists a closed and bounded subset of that is not compact.
Thus, by contraposition, we conclude that if every closed and bounded subset of a metric space is compact, then must be complete.
Would you like any further details or explanations?
5 Related Questions:
- How can we generalize the concept of compactness in non-metric spaces?
- Can a space be compact but not complete? Provide an example.
- What is the difference between sequential compactness and compactness in metric spaces?
- Does every compact metric space need to be bounded?
- Can you construct an example of a space that is bounded but not compact?
Tip:
In metric spaces, compactness and completeness are closely related, but remember that compactness implies completeness and boundedness, while completeness alone does not imply compactness.
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Math Problem Analysis
Mathematical Concepts
Metric Spaces
Compactness
Completeness
Boundedness
Cauchy Sequences
Formulas
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Theorems
Contrapositive Proof
Compactness Theorem
Completeness Theorem
Suitable Grade Level
Undergraduate Math (Advanced)
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