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 MM is complete if every closed and bounded subset of the metric space MM is compact. Here's a proof by contrapositive:

Contrapositive Statement:

We will prove the contrapositive of the original statement: If MM is not complete, then there exists a closed and bounded subset of MM that is not compact.

Proof:

  1. Assume MM is not complete. This means there exists a Cauchy sequence in MM that does not converge to a point in MM.

  2. Let {xn}\{x_n\} be a Cauchy sequence in MM that does not converge in MM. Since {xn}\{x_n\} is a Cauchy sequence, it is bounded (Cauchy sequences are always bounded in metric spaces).

  3. Now, consider the set A={xn:nN}lim supnxnA = \{x_n : n \in \mathbb{N}\} \cup \limsup_{n \to \infty} x_n. This set AA is closed because any limit point (if it exists) of the sequence must belong to AA, and all points xnx_n are in AA. However, since the sequence does not converge in MM, no limit point actually exists in AA, and hence AA does not have any accumulation point that lies in MM.

  4. Thus, the set AA is closed and bounded but not compact. In a metric space, compactness requires that every sequence has a convergent subsequence, but since the sequence {xn}\{x_n\} does not converge, there is no subsequence that converges to a point in AA.

  5. Therefore, AA is a closed and bounded subset of MM that is not compact.

This proves the contrapositive: If MM is not complete, there exists a closed and bounded subset of MM that is not compact.

Thus, by contraposition, we conclude that if every closed and bounded subset of a metric space MM is compact, then MM must be complete.


Would you like any further details or explanations?

5 Related Questions:

  1. How can we generalize the concept of compactness in non-metric spaces?
  2. Can a space be compact but not complete? Provide an example.
  3. What is the difference between sequential compactness and compactness in metric spaces?
  4. Does every compact metric space need to be bounded?
  5. 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)