> that definition will surely fall apart In what sense could that even conceivably be a true statement? I could define compact sets as sets which make me thin

> that definition will surely fall apart In what sense could that even conceivably be a true statement? I could define compact sets as sets which make me think of the color purple and it would be a completely valid definition. Nothing will fall apart. I think you are vaguely referring to the fact that proofs concerning compact sets in general topological spaces are not valid if you replace compact sets with bounded and closed sets, but that's just confusing the term "compact set" with what it refers to; the proofs aren't about the term but about the thing, which doesn't change. If "compact set" meant "bounded and closed" and, say, "cover-compact set" referred to sets where (ahem) every open cover has a finite subcover, then the concept of "cover-compact" will have certain general proofs associated to it and the term "compact" will have a different collection of general proofs associated to it. So in some ways you're making the same mistake the author did: carefully selecting a certain way to generalize a concept (real -> topological) and a certain ill-conceived claim about the original concept (compact=bounded and closed), and then acting all surprised when, oops!, the claim doesn't generalize.