I am looking for books with set theory and logic that is sufficient to understand mathematical analysis. I guess another question might be if there even exists such a book.
There are basically two problems I have seen in real analysis that requires set theory. They often create very big sets, but in set theory you can’t just create sets, you have to know why it is a set, in order to not get a paradox?
The second thing from set theory that is often used is the axiom of choice and zorn’s lemma.
Are there more things from set theory that is used in real analysis?(and also functional analysis)(apart from the operations of unions, intersections etc..)
Are there any books that gives a good(and hopefully easy) introduction to all that is needed of set-theory in mathematical analysis?
Try the first chapter of Topology by Munkres.
Two very standard texts on set theory are
Introduction to Set Theory by Hrbacek & Jech.
This book approaches the subject informally(not much formal logic) and has a good range of topics.
Also there is
Elements of Set Theory by Enderton.
This book requires some familiarity with formal logic and so it a bit more rigorous than Hrbacek & Jech. It doesn’t cover quite as many topics as the first book, but does cover anything you would need for real analysis.