4. Topology

Open sets, compactness, completeness — the topology of the real line, the Baire category theorem, Cantor sets.

• Open and Closed Sets — A set $U$ is open when every point is an interior point — each point can be wrapped in an $\varepsilon$-ball. Move the point $x$ and the radius $\varepsilon$ on the line and watch whether the ball stays inside.
• Closure and Boundedness — Drawing a bar over $A$ gives the closure $\overline{A}$ — the set with its escaped limit points pulled back in. Follow the round-song: closed ⟺ contains its limit points ⟺ equals its closure, and closed + bounded ⟺ compact (Heine-Borel).
• Compact Sets — Compact $\iff$ closed + bounded (Heine–Borel): $[0,1]$ is covered by the first 6 intervals of its open cover, while $(0,1]$ fails at the gap near 0 and $[0,\infty)$ fails at the infinite tail. Sequential compactness via Bolzano–Weierstrass is proved in full; the open-cover definition's equivalence is shown at the idea level.
• Metric Spaces — Axiomatizing distance generalizes the $\varepsilon$-ball — the shape of an open ball $B(c, r)$ depends on the metric $d$. Compare the Euclidean disk, the max-norm square, and the extreme discrete metric.
• Contraction Mapping Principle — A contraction $d(Tx,Ty) \le q\,d(x,y)$ ($q<1$) on a complete metric space has a unique fixed point — the Picard iterates $x_{n+1} = Tx_n$ tighten inside the error ball $\frac{q^n}{1-q}|x_1-x_0|$. $T(x) = x/2$ on $(0,1]$ shows why completeness is essential: a contraction with no fixed point.
• Baire Category Theorem — A complete metric space cannot be a countable union of nowhere dense sets — that is the Baire Category Theorem. Dodging the rationals of $[0, 1]$ one by one with nested closed intervals $I_k$ leaves a single irrational point in the intersection, whose lengths shrink to $0$.
• Baire Class 1 Functions — Pointwise limits of continuous functions — Baire class 1. Thomae's function $T(p/q) = 1/q$ (reduced), $T(x) = 0$ (irrational) is the pointwise limit of sums $f_n$ of disjoint tents. $T$ is continuous at irrationals, discontinuous at rationals — a living example of Baire's theorem that the continuity points contain a dense $G_\delta$.
• Length Zero, Points Uncountable — Deleting middle thirds drives the total length $\left(\frac{2}{3}\right)^n\to 0$, yet uncountably many points remain. Uncountability is stated at the idea level only.
• The Fat Cantor Set — What if we dig out less? Removing only length $4^{-n}$ per interval leaves a set that is compact, nowhere dense and perfect — yet keeps total length $\frac12$. Measure and topological size are independent notions.
• The Cantor Function — The Devil's staircase: continuous and nondecreasing from $0$ to $1$, yet constant on every complementary interval, so $f' = 0$ on a set of total length $1$. $f(1) - f(0) = 1 \neq \int_0^1 f'$ — why the FTC needs more than differentiability almost everywhere.