I was studying 2nd chapter of baby Rudin. And after problem 23 they introduced the concept of seperable space. About which I don't know any thing. I can solve problems after 23 by seeing hints.. but is it worth it?? I mean should I study separability from any toplology books and after knowing the concept fully I will try to solve it or should I move on to next chapter and keep it for later.
2026-04-04 20:56:25.1775336185
Advice on rudin
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Many proofs in real analysis hinge on special topological properties of the reals, such as Baire Categories, the Heine-Borel Theorem, existence of $\sup / \inf$ for bounded sets, etc. In a perfect world, every student of analysis would have a perfect understanding of topology and every student of topology would have a perfect understanding of real analysis. Alas, we all have to start somewhere.
One way to view your current conundrum is to put yourself in Rudin's shoes: you're trying to write a book on real analysis, but you need to borrow a lot of ideas from topology. How do you include the necessary material without having your real analysis book double in length and become a topology & real analysis book that would require two semesters?
This is not the last time you will ever come across this while learning mathematics. As such, my typical approach is to look at the following questions:
Is my understanding of this topic necessary for learning what I'm after? Am I currently trying to learn real analysis, topology, or mathematics? E.g., is your understanding of "separable" sufficient for learning the following bits of real analysis?
If yes, then safely read ahead, but be prepared to go back or consult other sources if you're getting lost. E.g., if you can follow the proofs and understand why it's necessary to have a separable space, you can probably put it on the back burner for now.
If no, then go consult another text on the relevant topic for an exposition that is strictly about said topic. E.g., if Rudin's use of separable spaces in the following sections seems foreign, go spend some time reading a topology textbook.