Exercises on Topology: Part 2

Nine more exercises, proved in full and animated step by step. We build a world where every distance equals one and watch open, closed, and compact change meaning — every subset clopen, compact means finite. We audit five candidate distance functions and unmask three impostors with live counterexamples, then prove a set compact with bare hands, straight from the definition: one cover set swallows the infinite tail, finitely many sets catch the stragglers. Then the theorem-breakers: a compact set with countably many limit points, the canonical cover of (0,1) with no finite subcover, nested closed or bounded sets with empty intersection, and a set of rationals that is open, closed, and bounded — yet not compact. Heine–Borel is a theorem about ℝᵏ, and we watch it die inside ℚ. The finale: the set of numbers whose digits are all 4s and 7s — uncountable, nowhere dense, compact, perfect, a true cousin of the Cantor set — and from it, a perfect set containing no rational number at all.