Question

Prove that every compact set is complete. Disprove the converse by examples.

Short answer

Every compact set is complete, but not every complete set is compact, as \\(\mathbb{R}\\) shows.

Step-by-step solution

Understanding Compactness

A set is called compact if every open cover has a finite subcover. This means that from any collection of open sets that completely cover the set, we can select a finite number that still cover the entire set.

Understanding Completeness

A set is complete if every Cauchy sequence within the set converges to a limit in the set. A Cauchy sequence is one where the terms get arbitrarily close as the sequence progresses.

Prove Compact implies Complete

Suppose we have a compact set. Take any Cauchy sequence in this set. By the property of compactness in Euclidean spaces, every sequence has a convergent subsequence. This subsequence converges to a limit within the compact set. Since the original sequence is Cauchy, it converges to the same limit within the set. Thus, the set is complete.

Disprove Complete implies Compact

Find a complete set that is not compact. Consider the set of real numbers \((-\infty, \infty)\). This set is complete, as every Cauchy sequence of real numbers has a limit that is also a real number. However, it is not compact because it does not have a finite subcover for open intervals that cover \(-\infty\) to \(+\infty\), such as \(\{(n, n+2) \mid n \in \mathbb{Z}\}\).

Key concepts

Completeness

Completeness is a fundamental concept often discussed in the realm of mathematics, specifically in analysis and topology. Essentially, a set is termed complete if every Cauchy sequence within it converges to a limit that also belongs to the set.
In simpler terms, imagine a sequence of numbers where each number is getting progressively closer to another. If this sequence finally lands, so to speak, at a number that is still a part of the original set, the set is complete.

• For example, the set of real numbers is complete because sequences of real numbers that "settle down" do so within the reals.
• Conversely, the set of rational numbers isn't complete, as Cauchy sequences of rationals, like \(\sqrt{2}\), don't always yield a rational limit.

Cauchy Sequences

A Cauchy sequence is a sequence where the numbers get closer to each other as you progress through the order of the sequence.
In formal terms, a sequence \((x_n)\) is Cauchy if for every positive number \( \varepsilon \), there’s a point beyond which each term of the sequence is within \( \varepsilon \) of each other.
This means for any degree of closeness, no matter how tiny, we can find a point in the sequence after which all further terms are incredibly close to one another.

• This behavior suggests that the sequence should "converge," although this isn't guaranteed unless additionally, the set is complete.
• Any sequence that eventually "levels out" and hits a point will behave like a Cauchy sequence.

Open Cover

The concept of an open cover is central in many areas of analysis and is particularly pivotal when discussing compactness. An open cover for a set is essentially a collection of open sets that together "cover" or encompass every point in the set.
Imagine placing infinite sheets of graph paper (open sets) precisely so that they cover every point of an object (the set).

• For each point in your set, at least one of these graph papers will contain it.
• A finite subcover is a finite selection from this collection that still manages to cover the entire set, illustrating its compactness.

This property is what allows a compact set to have a neatly packaged characterization, notably with finite resources, in terms of covering.

Convergent Subsequence

Subsequences are composed from sequences by picking some terms but still preserving their order. A convergent subsequence is one that approaches a specific point or limit.
For instance, if you take every second term of a sequence, you've created a subsequence. If this subsequence homes in on a particular value, it's convergent.
Convergent subsequences are particularly important when examining compact sets because they help demonstrate why compact sets are also complete.

• For any sequence within a compact set, there is always at least one subsequence which converges to a point that lies within the same set.
• This property helps connect compactness to completeness by showing how Cauchy sequences eventually converge inside the set.