Heine-Borel Theorem

A set K is compact if every open cover of K admits a finite subcover. An open cover is a collection of open sets whose union contains K. Finite subcover means you can throw away all but finitely many of those sets and still cover K. The definition is purely topological — no metric required — and captures a robust generalization of "finite". Compact sets behave like finite sets in many proofs: any continuous function attains its maximum, any continuous function-valued sequence has a convergent subsequence in the right setting, and the image of a compact set under a continuous map is compact.

Both conditions are needed. Bounded alone fails: (0, 1) is bounded but not closed; the cover {(1/(n+1), 1) : n ≥ 1} has no finite subcover because each set misses a tail near 0. Closed alone fails: ℝ is closed but unbounded; the cover {(−n, n) : n ≥ 1} has no finite subcover. Closed and bounded together force every sequence in K to have a convergent subsequence (Bolzano-Weierstrass), with limit inside K (closedness), which is sequential compactness — equivalent to compactness in ℝⁿ.

Riesz's lemma (1918): the closed unit ball of an infinite-dimensional normed space is never compact. In ℓ² (square-summable sequences), the standard basis vectors {eₙ} satisfy ||eₙ|| = 1 (so they sit in the closed unit ball, which is closed and bounded) yet ||eₘ − eₙ|| = √2 for m ≠ n, so no subsequence is Cauchy and no subsequence converges. The phenomenon is that infinite dimensions provide enough "room" for points to spread out without piling up. Compactness in infinite dimensions requires extra restrictions — equicontinuity (Arzelà-Ascoli) for function spaces, weak topology in functional analysis.

EVT: any continuous function f : K → ℝ on a compact set K attains its supremum and infimum on K. Proof: the image f(K) is compact (continuous image of compact is compact), hence by Heine-Borel closed and bounded in ℝ, so it has both a max and a min, both in f(K). On non-compact sets EVT fails: x ↦ x on (0, 1) has no max, x ↦ 1/x on (0, 1) has no max either. EVT is the basis of all of optimization theory: any continuous loss function on a compact constraint set is minimized.

Heine-Cantor theorem: any continuous function on a compact set is uniformly continuous. Pointwise continuity gives one δ per point per ε; uniform continuity demands a single δ that works everywhere given ε. Compactness lets you cover the domain with finitely many δ-balls; the minimum of finitely many positives is positive — that's the uniform δ. This is why ∫_a^b f(x) dx makes sense for any continuous f : [a, b] → ℝ — the partitions yield Riemann sums whose oscillation goes to zero uniformly.

Yes — in any metric space, compactness (every open cover has a finite subcover), sequential compactness (every sequence has a convergent subsequence), and being complete-and-totally-bounded are all equivalent. In general topological spaces these split: compactness is the topological definition; sequential compactness is weaker; the Tychonoff theorem (a product of compact spaces is compact) requires open-cover compactness. For analysis problems set in ℝⁿ or in metric spaces, sequential compactness is the workhorse — it's how you extract convergent subsequences in approximation arguments.