Understanding Complete Metric Spaces and Their Closed Subsets

When discussing metric spaces, it's crucial to understand the concepts of completeness and closed subsets. A metric space is considered complete if every Cauchy sequence of points within the space converges to a limit that is also within the space. This article will explore the nuances of these concepts, clarifying why a metric space may not be complete even if it is closed. We will also delve into the specific example of rational numbers and the importance of compactness in metric spaces.

Complete Metric Spaces: Defining the Concept

A metric space (X) is said to be complete if every Cauchy sequence in (X) converges to a limit that is also in (X). This is a fundamental property that ensures the space has no "missing points".

Examples and Clarification: The Set of Rational Numbers

Let's consider the set of all rational numbers, denoted as (mathbb{Q}), with the standard metric. Despite (mathbb{Q}) being closed (as it contains all its limit points), it is not a complete metric space. An example of a sequence in (mathbb{Q}) that converges to a point not in (mathbb{Q}) is the sequence ({ frac{1}{n} }_{n in mathbb{N}}), which converges to (0). Since (0) is an irrational number, it is not included in (mathbb{Q}). Therefore, (mathbb{Q}) is not complete.

Compactness and Completeness

The compactness of a metric space is a stronger condition than completeness. If a metric space (X) is compact, it is also complete. This is because every sequence in a compact metric space has a convergent subsequence which must converge to a limit within the space, thus satisfying the completeness condition.

Definition of Closed Subsets and Limit Points

A subset (A) of a metric space (X) is considered closed if it contains all of its limit points. However, this condition is not sufficient to ensure completeness. For instance, consider the set of all integers with the standard metric, denoted as (mathbb{Z}). Since (mathbb{Z}) has no limit points (no point in (mathbb{Z}) is a limit point of (mathbb{Z})), it is trivially closed. However, (mathbb{Z}) is not complete, as sequences like ({ frac{1}{n} }_{n in mathbb{N}}) converge to (0), which is not in (mathbb{Z}).

Conditional Nature of Closed Subsets

The definition of a closed subset is conditional. It ensures that any sequence in (C subseteq X) with a limit in (X) must have that limit in (C). It does not guarantee that every sequence in (C) will converge, only that if it does converge, the limit must be in (C).

Conclusion

In summary, while a metric space (X) being closed means it contains all its limit points, it does not necessarily imply that every Cauchy sequence converges within (X). This is why (mathbb{Q}), for instance, is not complete despite being closed. The completeness of a metric space is a stronger condition, and compactness ensures both completeness and the property of being closed.