What is compact metric space?

Example 4.17 (Heine’s theorem, with and without compactness)

On $\intcc{0}{1}$, the function $x \mapsto x^2$ is uniformly continuous — Heine says so with no computation, but the direct estimate is instructive:

so $\delta = \varepsilon/2$ works for every point at once. On $\R$ the same function is not uniformly continuous: with $x_n = n$ and $y_n = n + \frac1n$, the gap $\abs{x_n - y_n} = \frac1n \to 0$ while $\abs{x_n^2 - y_n^2} = 2 + \frac1{n^2} \geq 2$: no single $\delta$ serves $\varepsilon = 1$. The mechanism is visible: the local Lipschitz constant $\abs{x + y}$ is bounded on a compact and unbounded on $\R$ — Heine’s theorem is exactly the statement that compactness caps such local constants uniformly.

Example 4.19 (Distances between sets: compactness earns its keep)

Let $K$ be compact, $F$ closed, $K \cap F = \emptyset$ in a metric space. Then

the function $x \mapsto d(x, F)$ is continuous (Exercise 4.11) and positive on $K$ ($d(x, F) = 0$ would put $x \in \overline F = F$), so it attains a positive minimum on the compact $K$ (Theorem 4.16 (3)). Compactness is not decorative: for two closed sets the infimum can vanish without being attained — in $\R^2$, the hyperbola $F_1 = \{xy = 1\}$ and the axis $F_2 = \{y = 0\}$ are disjoint closed sets with $d(F_1, F_2) = 0$ (the points $(n, \frac1n)$ approach the axis). Escape to infinity is exactly what compactness forbids.

Example 4.21 (An $\varepsilon$-net, counted)

Total boundedness (from the proof of Theorem 4.20) is very concrete on $\intcc{0}{1}$: for $\varepsilon > 0$, the $\lceil \frac{1}{2\varepsilon}\rceil$ balls centered at $\varepsilon, 3\varepsilon, 5\varepsilon, \dots$ of radius $\varepsilon$ cover it — about $\frac1{2\varepsilon}$ balls, and no cover can do with fewer than $\frac{1}{2\varepsilon}$ of them (each ball covers length at most $2\varepsilon$). In $\intcc01^2$ the count squares to order $\varepsilon^{-2}$: covering numbers grow like $\varepsilon^{-d}$ in dimension $d$ — a quantitative face of compactness, and the reason the infinite-dimensional unit balls of Chapter 5 (where no finite $\frac13$-net exists at all) cannot be compact.