Compactness and extreme value theorems

An open cover of K is a collection of ambient open sets whose union contains K. Compactness says that some finite subcollection still covers K. The original cover can be infinite, its members can overlap, and a finite subcover need not be unique or minimal. To prove a set is not compact, it is enough to give one open cover with no finite subcover. For (0,1), the relative open sets (1/n,1), with n at least two, cover the interval, but any finite selection misses sufficiently small positive points.

In metric spaces, compactness and sequential compactness are equivalent. One bridge is a Lebesgue number: an open cover of a compact metric space admits a positive radius such that every ball of that radius, relative to the space, fits inside some cover member. Combined with a finite net, this yields a finite subcover. The metric assumption matters when translating freely between sequences and covers; more general topological spaces need separate care. Here either formulation can be used, provided limits remain in K.

Let f map a compact space K continuously into a metric space Y. Pull an open cover of f(K) back through f. Continuity makes each inverse image open relative to K, and together these inverse images cover K. A finite subcover of K then supplies the corresponding finite collection covering f(K). This proof reveals why inverse images, rather than images, of open sets appear in the definition of continuity: inverse images preserve the local structure needed to transfer a cover argument.

A nonempty compact subset of the real line is bounded and contains its supremum and infimum. Therefore a continuous real-valued function on a nonempty compact domain attains both its minimum and its maximum. Alternatively, choose points whose values approach the infimum, extract a convergent subsequence in K, and use continuity to pass to the limiting value. This second proof distinguishes existence from computation: it certifies that a minimizer exists, without supplying an efficient way to find it or claiming that it is unique.

Removing compactness can lose attainment even when values stay bounded. On (0,1), the continuous function f(x)=x has infimum zero and supremum one but attains neither. Removing continuity can fail on a compact set: define f(0)=1 and f(x)=x for positive x in [0,1]. Its infimum is zero, but no point has value zero. These examples isolate separate failure mechanisms. A theorem's hypotheses are not decorative; they block specific counterexamples that can otherwise resemble well-behaved numerical problems.

In finite-dimensional Euclidean space, the Heine–Borel theorem says that closed bounded sets are compact. The finite-dimensional condition cannot be dropped. In general metric spaces use compactness itself, or completeness plus total boundedness, rather than closedness plus boundedness. A different existence strategy is coercivity: if a continuous real objective on Euclidean space tends to positive infinity when the input norm tends to infinity, a suitable compact sublevel set traps a minimizing sequence. That argument still uses compactness after constructing an appropriate bounded region.