Möbius Band as a Bundle
The example to carry everywhere. The Möbius band is a Fiber Bundle over $S^1$ with fiber an interval (or line): locally $U \times \mathbb{R}$, globally twisted. The cylinder $S^1 \times \mathbb{R}$ is its trivial sibling — locally identical, globally different.
Construction via transition data. Cover $S^1$ by two arcs $U_1, U_2$ overlapping in two disjoint pieces. Take Transition Functions $t = +1$ on one piece, $t = -1$ on the other (structure group $\mathbb{Z}_2 = O(1)$, or $\{\pm 1\} \subset GL(1,\mathbb{R})$). The $-1$ is the twist; no choice of trivializations removes it.
What it teaches, per concept. - Local Trivialization: both patches look like products; the twist is invisible locally. - Section of a Bundle: every global section (as a line bundle) must cross zero — a continuous "width-coordinate" function on the band must vanish somewhere. Nontriviality has observable consequences for sections. - Vector Bundle: the unique nontrivial real line bundle over $S^1$; classified by $H^1(S^1;\mathbb{Z}_2) = \mathbb{Z}_2$ — the simplest characteristic class (first Stiefel–Whitney).
Physics echo. A field that flips sign around a loop = a section of a Möbius-like bundle: antiperiodic boundary conditions, Berry phase of $\pi$. The Möbius band is the $\mathbb{Z}_2$ ancestor of the Dirac Monopole's $U(1)$ twisting.
Related: Fiber Bundle · Transition Functions · Hopf Fibration · Module 05 - Fiber Bundles - First Definitions