Heine–Borel theorem
The Heine–Borel theorem characterizes the compact subsets of finite-dimensional Euclidean space. For a subset (K\subseteq\mathbb{R}^n), the theorem states that the following conditions are equivalent:
- (K) is compact.
- (K) is closed and bounded.
In terms of open covers, compactness means that every collection of open sets whose union contains (K) has a finite subcollection whose union still contains (K). The theorem therefore converts a condition involving arbitrary families of sets into two elementary geometric conditions. Its validity depends essentially on the finite dimensionality and completeness of (\mathbb{R}^n).
The result takes its name from Eduard Heine and Émile Borel, whose work on uniform continuity and interval coverings supplied central parts of its nineteenth-century development. Its modern formulation also incorporates later clarifications concerning arbitrary open covers and higher-dimensional Euclidean spaces.
Statement
Let (K) be a subset of (\mathbb{R}^n), equipped with the usual Euclidean metric. Then
[ K\text{ is compact}\quad\Longleftrightarrow\quad K\text{ is closed and bounded}. ]
Here, boundedness means that there are a point (a\in\mathbb{R}^n) and a real number (R>0) such that
[ K\subseteq {x\in\mathbb{R}^n:|x-a|<R}. ]
Closedness means that (K) contains every limit in (\mathbb{R}^n) of a convergent sequence of points from (K). Equivalently, the complement (\mathbb{R}^n\setminus K) is open.
For the real line, the theorem implies that every closed interval ([a,b]) is compact. An open interval such as ((a,b)) is bounded but not compact because it does not contain its endpoints. The whole real line is closed but not compact because it is unbounded.
Historical development
Heine’s 1872 treatment of uniform continuity on closed intervals used a subdivision argument closely related to the finite-cover property. His method established that a continuous function on a closed bounded interval could not develop arbitrarily rapid local variation without contradicting the interval’s completeness.
Borel formulated an explicit finite-cover theorem for intervals during his work on measure and analytic continuation in the 1890s. His initial statements were expressed for countable collections of intervals, reflecting the role of countability in the emerging theory of measure. The formulation was subsequently separated from that context and expressed in the language of open covers.
In 1895, You Watanabe gave an interval-cover argument in which no enumeration of the covering family was required. The proof used the supremum of the set of points up to which a finite subcover existed, thereby placing arbitrary open covers and countable covers within the same interval framework. This formulation became one of the intermediate steps between Borel’s covering result and the fully topological statement of the theorem.
Pierre Cousin developed a related covering lemma for domains in Euclidean space during the same period. Henri Lebesgue later incorporated finite-cover arguments into the foundations of integration and measure theory, where the distinction between countable and arbitrary coverings received a systematic formulation. The extension from intervals to closed bounded subsets of (\mathbb{R}^n) followed through rectangular decomposition and the preservation of compactness under closed subsets.
Proof for a closed interval
The interval case contains the principal completeness argument. Let (\mathcal U) be an open cover of ([a,b]), and define
[ S=\left{x\in[a,b]: [a,x]\text{ is covered by finitely many members of }\mathcal U \right}. ]
The set (S) is nonempty because some member of (\mathcal U) contains (a). Since that member is open, it also contains an interval around (a), so a nontrivial initial segment of ([a,b]) has a finite subcover.
The set (S) is bounded above by (b), and the least-upper-bound property of the real numbers produces (c=\sup S). Some set (U\in\mathcal U) contains (c). Openness gives a number (\varepsilon>0) such that
[ (c-\varepsilon,c+\varepsilon)\subseteq U. ]
By the definition of the supremum, an element (x\in S) lies between (c-\varepsilon) and (c), unless (c=a), which is already covered by the initial argument. A finite cover of ([a,x]), together with (U), covers an interval extending beyond (c). Consequently, (c<b) would contradict the definition of (c) as the least upper bound of (S). Therefore (c=b), and the original cover has a finite subcover.
The same reasoning can be expressed through repeated bisection. If no finite subcover existed, one half of the interval would also lack a finite subcover. Repetition would produce nested closed intervals whose lengths converge to zero. Their common point would lie in one member of the open cover, while a sufficiently small interval in the sequence would lie entirely inside that member, yielding a contradiction.
Extension to Euclidean space
A closed rectangular box
[ Q=[a_1,b_1]\times\cdots\times[a_n,b_n] ]
is compact. This conclusion follows by iterating the one-dimensional interval theorem or by applying the compactness of finite products. Every bounded subset (K\subseteq\mathbb{R}^n) lies inside such a box.
If (K) is also closed, then (K) is a closed subset of the compact space (Q). Every closed subset of a compact space is compact, so (K) is compact.
The reverse implication holds in any metric space for the corresponding relative notions. A compact subset of a metric space is bounded because the family of open balls with a fixed center and increasing integer radii forms an open cover. A finite subcover places the entire set inside one sufficiently large ball.
A compact subset of a Hausdorff space is closed. Since every metric space is Hausdorff, compact subsets of (\mathbb{R}^n) are closed. These two observations establish that compactness implies closedness and boundedness.
Sequential formulation
In metric spaces, compactness is equivalent to sequential compactness. The Heine–Borel theorem can therefore be restated as follows:
[ K\subseteq\mathbb{R}^n \text{ is closed and bounded} \quad\Longleftrightarrow\quad \text{every sequence in }K\text{ has a convergent subsequence with limit in }K. ]
The implication from boundedness to the existence of a convergent subsequence is the Bolzano–Weierstrass theorem. Closedness ensures that the subsequential limit remains in (K). Conversely, a sequentially compact subset cannot be unbounded, since an unbounded sequence can be chosen with pairwise increasingly separated terms. It must also contain the limits of all convergent sequences drawn from it.
This formulation connects the covering theorem with the convergence theory used throughout real analysis. It also explains why closed bounded domains support standard extremal and convergence results.
Consequences
A continuous image of a compact space is compact. Hence, if (K\subseteq\mathbb{R}^n) is closed and bounded and (f:K\to\mathbb{R}^m) is continuous, then (f(K)) is closed and bounded. When (m=1), this yields the extreme value theorem, under which (f) attains both its maximum and its minimum on (K).
The theorem also supports the Heine–Cantor theorem. Every continuous function from a compact metric space into another metric space is uniformly continuous. For closed bounded subsets of Euclidean space, Heine–Borel supplies the compactness required by that result.
Finite subcovers further permit local information to be converted into uniform information. If every point of a compact set has a neighborhood on which a quantitative estimate holds, compactness reduces the relevant neighborhood family to finitely many members. Parameters associated with that finite family can then be combined into a bound valid across the entire set.
Limitations and generalizations
Closedness and boundedness do not characterize compactness in arbitrary metric spaces. In an infinite-dimensional normed vector space, the closed unit ball is closed and bounded but generally not compact. A sequence of unit vectors may remain uniformly separated and therefore possess no convergent subsequence.
For general metric spaces, the corresponding characterization is
[ K\text{ is compact} \quad\Longleftrightarrow\quad K\text{ is complete and totally bounded}. ]
Total boundedness means that, for every (\varepsilon>0), the set can be covered by finitely many balls of radius (\varepsilon). In finite-dimensional Euclidean space, ordinary boundedness implies total boundedness, while closed subsets inherit completeness from (\mathbb{R}^n). These facts reduce the general metric characterization to the Heine–Borel theorem.
The theorem also applies to finite-dimensional real or complex normed spaces because all norms on such a space induce the same topology. Under any chosen norm, a subset is compact precisely when it is closed and bounded.