Best book on general topology for functional analysis

Question

I study functional analysis. From time to time I find that the basics of many concepts lie on the concepts of topology.

I am familiar with basic concepts of topology. When I go deeper then wild things start.

I am not interested in topology for the sake of topology. I need it for a deep understanding of functional analysis.

Question 1: Is there any good book on topology for my purpose?

Question 2: If I really want to be good in functional analysis do I need study the whole topology or I can be satisfied by the standard topology?

Question 3: My question shows my lack of understanding and immaturity in mathematics. I should not be distracted from the main curriculum. Yes or no?

(My area of interest is harmonic analisys and operator theory. I am newbie).

Answer 1

1. Any standard book on point-set topology would work. Munkres is the canonical recommendation, and is good.
2. What do you mean the 'whole topology' and the 'standard topology'? Regardless, for functional analysis you'll probably just need regular point-set topology, nothing algebraic. For real analysis (i.e., what you studied before functional), you probably needed very little topology--just some ideas about open and closed sets. But for functional analysis, you want to understand very well the product topology and Tychonoff's theorem, so that the weak and weak* topologies make sense.
3. I'm not entirely sure how to parse this question. I will say that it's a bad idea to study functional analysis if you don't understand very well topology and real analysis (including measure theory) very well already. Particularly, you should be able to do the problems in, say, the first half of Rudin's Real and Complex Analysis. (Without a solid background in real analysis, functional analysis will make little sense and you won't have the right examples. If you go in without understanding $L^p$ spaces and how to use them, then a lot of the motivation for functional analysis will be lost. Rudin is nice for this since his book includes the beginnings of functional analysis--basically everything you can do without getting into hard material: Analyzing basic properties of Banach spaces with attention paid to $L^p$, and using Hilbert spaces to solve certain optimization problems and develop the theory of Fourier series. Going for an abstract treatment of functional analysis without knowing these concrete motivations is a very big mistake.) Also, know finite-dimensional linear algebra down cold. You should be able to prove the spectral theorem in finite-dimensions, know about canonical forms of matrices, etc. like the back of your hand. Linear algebra is useful in all branches of mathematics, and a large part of functional analysis is trying to extend the tools of finite-dimensional linear algebra to make sense in infinite-dimensions. If you don't understand very well the finite-dimensional case, a lot of the motivation for functional analysis will be lost.