There are standard texts on the various topics listed in the contents. There are online texts on these topics as well as free to download texts. To learn everything, get a math degree or self-study the curriculum of one. The prerequisite is basically calculus.
The thing about this text is it seems like a more or less random list of theorems - it lists one theory, I think, Zorn's lemma, under topology where Zorn's lemma is more a set-theory-link-to-topology (one of the guises of the Axiom of choice, really). Which to say it has next-to-nothing on point-set topology proper. Other theory listings seem just as random. So I couldn't what order you'd follow trying to learn everything specifically here.
The thing about this text is it seems like a more or less random list of theorems - it lists one theory, I think, Zorn's lemma, under topology where Zorn's lemma is more a set-theory-link-to-topology (one of the guises of the Axiom of choice, really). Which to say it has next-to-nothing on point-set topology proper. Other theory listings seem just as random. So I couldn't what order you'd follow trying to learn everything specifically here.