I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components do). While I might be missing steps (I based the proofs off some optional exercises in a textbook, but the proof of the Tychonoff theorem was mostly my own), it still seemed much cleaner and certainly more than other proofs of the theorem I've seen, specifically the ones based on Zorn's Lemma/the Hausdorff Maximal Principle.
My question is why more authors don't use this method of proof. In all (two of) the topology books I've read, either the author didn't prove the theorem or used the other approach, and I'm curious why.
More generally, I'm wondering why more topology books don't talk primarily about nets and leave sequences as a special kind of net to be used in counterexamples. While there's obviously a hurdle in that you have to discuss directed sets (which are more abstract), it seems like nets would make a lot of the results about compactness, and their proofs, much cleaner.