Point-set topology serves as indoctrination in set theory, in disguise, because it looks like geometry and analysis at first. What it is doing is reformulating obvious elementary concepts from real analysis in set language, and then imposing extra non-obvious set-theoretic conceptions in a language that makes them look natural and seductive (Tychonov's theorem for real-products of spaces example, in lieu of real-number choice). The point-set aspects are totally sterile unless you understand the nuances of set theory well enough to know what they really mean, and that requires learning countable models of set theory, logic, forcing, so that you can answer the point-set questions three ways "yes, no, it depends on the set-theory model", because the most natural questions in point-set topology are set-theoretic questions about uncountable sets, and so have "it depends on the set-theory model" as the best answer.