How to start reading topology? I've recently completed my high school. I want to read topology. How should I begin and from where should I begin? 
 A: I don't think this is very reasonable to begin to study topology just after high school, as you will feel you need a bit more background. If you feel comfortable with calculus, I would advise the wonderful Introduction to topological manifolds by John Lee, which is extremely clear and well-written, and contains very good material.
A bit different but still good is Topology, by James Munkres. It is more "pure topology" oriented but contains many details and many interesting counter-examples, exercices and example. The first chapter is also giving you necessary background in set theory which can be useful. Good luck !
A: Basic Topology by Armstrong.  For me it was an awesome experience to read it. very inspiring!
A: Not really a book recommendation, but...
I know at least one math PhD whose first college math course was point set topology. She caught the math bug because of this class. It's an odd thing to study first, but it can be done - introductory point set topology can be a nice introduction to mathematical definitions and proofs, and doesn't require much knowledge other than some naive set theory.
The reason it might be "odd" is that a lot of the definitions will seem arbitrary and unmotivated. The topic also likely won't have much to do with the interesting part that has drawn you to the subject in the first place. Those require getting to "algebraic topology," which really does require more knowledge of undergraduate math.
I can't make a good book recommendation for "point set topology," (often called "general topology") but there are books out there, and plenty of information online.
