A set K in ℝ is compact if every open cover has a finite sub-cover. The Heine-Borel Theorem states that a set is compact if and only if it is closed and bounded. Compact sets have many excellent properties: continuous images are compact, and continuous functions attain extrema on compact sets.
You already know the distinction between open and closed sets on the real line, and you may know the Bolzano-Weierstrass theorem — every bounded sequence has a convergent subsequence. Compactness is the structural property that explains *why* results like Bolzano-Weierstrass hold, and it does so through a deceptively elegant definition: a set K is compact if every open cover of K has a finite subcover. An open cover is any collection of open sets whose union contains K; the definition demands that no matter how many (even infinitely many) open sets you choose to cover K, you can always discard all but finitely many and still cover K.
The definition sounds abstract until you see it fail. Consider the open interval (0, 1). Cover it with the open sets (1/n, 1) for n = 1, 2, 3, … Every point in (0, 1) is eventually in one of these sets, so this is a valid cover. But no finite subcollection works — any finite sublist misses a neighborhood of 0. This is why (0, 1) is not compact: it's bounded but not closed. Now try the same trick on [0, 1]. You can't: any open cover of [0, 1] must include a set covering 0 and a set covering 1, and once those are in, finitely many more sets suffice. Closedness is what "traps" the boundary and prevents the cover from sliding away to infinity.
The Heine-Borel Theorem makes this precise: in ℝⁿ, a set is compact *if and only if* it is closed and bounded. This is a characterization unique to Euclidean space — the equivalence fails in general metric spaces. Boundedness ensures you can't escape to infinity; closedness ensures limit points are included. Together they make every open cover reducible to a finite one. The real payoff comes from what compact sets guarantee: if f is continuous on a compact set K, then f(K) is also compact (hence closed and bounded), and f attains its maximum and minimum on K. The Extreme Value Theorem you know from calculus is a corollary — it works *because* [a, b] is compact.
The Bolzano-Weierstrass theorem is the sequential counterpart: every sequence in a compact set has a convergent subsequence whose limit is also in the set. This sequential compactness characterization is equivalent to open-cover compactness in ℝⁿ, and it's often easier to apply in proofs. Think of compactness as a finiteness property dressed in topological clothing: it limits how "infinite" the structure of a set can be, forcing any description of its neighborhoods to reduce to something manageable. That finiteness is what makes compact sets the preferred domain for analysis — it is much easier to prove theorems when you cannot escape to infinity or sneak out through a missing boundary point.