What is Limits, continuity?

Example 4.2

$\R$ with $\abs{x - y}$; $\R^n$ with any of

the set $C(\intcc{a}{b})$ of continuous functions with the sup distance $d_\infty(f, g) = \sup_{\intcc{a}{b}} \abs{f - g}$ (finite: $f - g$ is bounded); any set with the discrete distance ($d(x,y) = 1$ for $x \neq y$). Distances coming from norms are the subject of Chapter 5.

Example 4.10 (Open and closed sets recognized by continuity)

The global characterization (Theorem 4.6) is the everyday tool for topological bookkeeping. In $\R^2$: the set $\{(x, y) : x^2 + y^2 < 1,\ y > x^3\}$ is open — it is $g^{-1}(\intoo{-\infty}{1}) \cap h^{-1}(\intoo{0}{+\infty})$ for the continuous $g(x,y) = x^2 + y^2$ and $h(x, y) = y - x^3$, an intersection of two open preimages. In $\bigl(C(\intcc01), d_\infty\bigr)$: the set of functions with $f(0) = f(1)$ and $\int_0^1 f = 0$ is closed — the preimage of $\{(0,0)\}$ under the continuous map $f \mapsto \bigl(f(0) - f(1),\ \int_0^1 f\bigr)$ into $\R^2$ (each coordinate is $1$-Lipschitz, as in Exercise 4.3). The method never draws a picture: exhibit a continuous map, read the set as a preimage, quote the theorem.

Example 4.11 (A closed set defined by infinitely many conditions)

In $\bigl(C(\intcc01), d_\infty\bigr)$, the set

of $1$-Lipschitz functions is closed, although it is cut out by uncountably many conditions: for each fixed pair $(x, y)$, the map $f \mapsto \abs{f(x) - f(y)} - \abs{x - y}$ is continuous (evaluations are $1$-Lipschitz), so each single condition defines a closed set, and $L$ is the intersection of this family — an arbitrary intersection of closed sets is closed. The same template certifies closedness for monotone functions, convex functions, functions bounded by a fixed $g$: uniform limits inherit every property expressible as a family of closed pointwise constraints. What uniform limits do not automatically inherit — differentiability, for one — is exactly what Chapter 10 must labour for.