The question, as stated in the title, is on the relationship between the two topics. A more specific one is that of "What is the motivation for introducing (Point Set) Topology into Analysis".
The question raised when I was trying to learn Mathematical Analysis and I realised that some books introduced the language of topology early and use them for the rest of the book (e.g. baby Rudin) while others not so (Tao's Analysis I. Metric and other topology terms only appear in the Analysis II, not I.). It seems to me that some theorems can be proved without using the language of topology. Does topology simplify the proofs? Or it is just that a lot of things later on in Analysis cannot be proved without topology, so they felt might as well start with them from the beginning?
I am also wondering about the history of when concepts of topology were introduced into analysis, and the motivation behind those introductions.
In addition, I heard the connection between the two topics becomes tighter when we study Functional Analysis. Do you mind to elaborate a bit on that remark?
