Compactness in Metric Spaces

Compactness is one of the most powerful and frequently applied concepts in analysis and topology. It generalizes the intuitive idea of a set being "closed and bounded" from Euclidean space to the abstract setting of metric spaces, providing a framework for proving the existence of objects like maxima, minima, and convergent subsequences. Understanding its equivalent characterizations unlocks the ability to switch between different powerful perspectives depending on the problem at hand.

The Definition: Compactness via Open Covers

The most fundamental definition of compactness is topological. An open cover of a set $K$ in a metric space $(X,d)$ is a collection $\{U_{\alpha }\}$ of open subsets of $X$ such that $K\subseteq {\bigcup }_{\alpha }{U}_{\alpha }$. A finite subcover is a finite sub-collection of these sets that still covers all of $K$.

A set $K$ is compact if every open cover of $K$ has a finite subcover. At first glance, this property—that you can always reduce an infinite cover to a finite one—seems technical. Its power lies in its ability to transfer local properties to global ones. If every point in $K$ has a neighborhood with some property (like being within an open set of a cover), compactness guarantees you only need to check a finite number of those neighborhoods to understand the entire set $K$.

Consider the interval $(0,1)$ in $\mathbb{R}$ with the standard metric. The collection of open intervals $\{(1/n,1){\}}_{n=2}^{\infty }$ is an open cover. However, any finite subcollection, say $\{(1/n_1,1),...,(1/n_k,1)\}$, has a largest $N=\max \{n_1,...,n_k\}$. This finite union is just $(1/N,1)$, which does not cover points near $0$. Since we found an open cover with no finite subcover, $(0,1)$ is not compact. In contrast, the closed interval $[0,1]$ is compact (this is the Heine-Borel Theorem for $\mathbb{R}$).

Sequential Compactness

In metric spaces, a more intuitive and often more usable characterization exists. A set $K$ is sequentially compact if every sequence in $K$ has a subsequence that converges to a limit point *also in $K$*. This directly captures the idea of "no escape to infinity" within the set.

For example, the sequence $x_n=n$ in $\mathbb{R}$ has no convergent subsequence, so $\mathbb{R}$ is not sequentially compact. The sequence $x_n=1/n$ in $(0,1]$ has a subsequence converging to $0$, but $0\notin (0,1]$, so $(0,1]$ fails sequential compactness as well. In the compact set $[0,1]$, the sequence $1/n$ converges to $0$, which is in the set. A more complex sequence like $\sin (n)$ in $[-1,1]$ is bounded; the Bolzano-Weierstrass theorem (which holds in ${\mathbb{R}}^n$) guarantees it has a convergent subsequence, illustrating the principle.

Theorem: In a metric space $(X,d)$, a set $K$ is compact if and only if it is sequentially compact.

This equivalence is a cornerstone of metric space analysis. The proof direction (compact $\Rightarrow$ sequentially compact) often uses a contradiction argument: if a sequence had no convergent subsequence, you could construct an open cover of balls that miss infinitely many terms, which would have no finite subcover. The converse direction typically uses the concept of total boundedness, which leads us to the next equivalent characterization.

Total Boundedness and Completeness

Two other metric properties combine to give a third equivalent definition of compactness. A set $K$ is totally bounded if, for every $ϵ>0$, there exists a finite set of points $\{x_1,x_2,...,x_n\}\subseteq K$ such that $K$ is covered by the union of the open balls $B_{ϵ}(x_1),...,B_{ϵ}(x_n)$. This finite set is called an $ϵ$-net. Total boundedness is a stronger condition than ordinary boundedness; it means the space can be approximated by a finite number of points to any desired precision.

A metric space is complete if every Cauchy sequence converges to a point within the space. A subset is complete if it contains the limits of all its Cauchy sequences.

Theorem: In a metric space, a set $K$ is compact if and only if it is both totally bounded and complete.

This characterization is immensely practical. To prove a set is compact, you can often check these two properties separately. Total boundedness handles the "finiteness" aspect (preventing the set from being too spread out), while completeness handles the "closedness" aspect (ensuring no points are missing from the set). For instance, the set $\{1/n:n\in \mathbb{N}\}\cup \{0\}$ in $\mathbb{R}$ is totally bounded (it's contained in $[0,1]$) and complete (it contains all its limit points, notably $0$), hence it is compact.

Applications of Compactness

Compactness is a tool for proving existence. A classic application is in optimization: a continuous real-valued function $f$ on a compact metric space $K$ attains its maximum and minimum. The proof uses the open cover definition: the set of points where $f$ is less than some value is open, and compactness allows you to find a finite subcover that pins down the minimum value.

Another vital application is in establishing uniform continuity. If a function $f:K\to Y$ is continuous and $K$ is compact, then $f$ is uniformly continuous. The local estimates of continuity provided by the $ϵ-\delta$ definition can be patched together into a single, globally applicable $\delta$ using the finite subcover property of compactness.

These applications show why compactness is non-negotiable in advanced analysis. It turns qualitative local information into rigorous, quantitative global statements.

Common Pitfalls

1. Assuming "Bounded and Closed = Compact": This is only true in specific spaces, like ${\mathbb{R}}^n$ with the standard metric (the Heine-Borel Theorem). In general metric spaces, a set can be closed and bounded but not compact. For example, the closed unit ball in an infinite-dimensional Hilbert space is closed and bounded but not totally bounded, hence not compact.
2. Confusing Compactness with Completeness: A set can be complete but not compact (e.g., the real line $\mathbb{R}$ is complete but not totally bounded). Conversely, a subset of a metric space can be totally bounded but not complete (e.g., $(0,1]$ in $\mathbb{R}$ is totally bounded but not complete, as the Cauchy sequence $1/n$ does not converge to a point in the set).
3. Misidentifying the Limit Point in Sequential Compactness: It is not enough for a sequence in $K$ to have a convergent subsequence; the limit of that subsequence *must be in $K$*. This is why $(0,1]$ is not sequentially compact, even though $1/n$ has a convergent subsequence.
4. Overlooking the Finite Subcover Requirement: When proving a set is not compact, you must demonstrate a specific open cover that has no finite subcover. You cannot argue generally about the difficulty of finding subcovers; you must construct a counterexample.

Summary

• Compactness is fundamentally defined by the open cover property: every open cover has a finite subcover.
• In metric spaces, compactness is equivalent to sequential compactness (every sequence has a convergent subsequence with limit in the set).
• It is also equivalent to being both totally bounded (can be covered by finitely many $ϵ$-balls for any $ϵ>0$) and complete (contains all its Cauchy limits).
• Compactness is a crucial tool for proving existence theorems, such as the attainment of extrema by continuous functions and the guarantee of uniform continuity.
• The familiar "closed and bounded" condition only implies compactness in special spaces like ${\mathbb{R}}^n$, not in all metric spaces.