Exercises on Topology: Part Four

Part 4 of 4, and the payoff of the series: the hardest results, the strangest objects, and the deepest reason any of it holds together. We begin with compactness from the inside out — why a space in which every infinite subset has a limit point must already be compact — proved cleanly through the Lebesgue number lemma instead of chasing remainders down a countable subcover. Then we step deliberately off the edge, into a space that has the limit-point property yet fails to be compact. That forces us to meet the ordinals properly, so we build them from nothing in the smallest possible steps: counting past infinity to ω, the difference between a successor and a limit, climbing through ω·2, ω², and the whole countable tower, and finally the leap to the first uncountable ordinal — the space that breaks the pattern. From there, condensation points. In any uncountable set on the line, all but countably many points are surrounded by uncountably many others, which carves a perfect core out of every uncountable set and gives the Cantor–Bendixson decomposition of any closed set into a perfect part and a countable part. A measure-zero set whose condensation points fill the entire real line shows just how violent the idea can get. We then anatomize every open set on the line as an at most countable union of disjoint segments, built almost for free from a single equivalence relation. We close with the Baire category theorem in its dense-open form, and then the example I think matters most in the whole series: the rationals, taken as a space on their own, written as a countable union of thin closed points that nonetheless fill everything, with Baire's conclusion failing completely. We take this one very slowly and watch exactly why it breaks — a tower of nested rational intervals closing in on √2, a hole where a real number should sit. Completeness is not a technicality; it is the entire reason the theorem is true.