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I've recently completed my high school. I want to read topology. How should I begin and from where should I begin?

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  • $\begingroup$ While you wait for knowledgeable people to answer, I recommend watching this video by 3blue1brown, about an unsolved problem in topology and the elegant solution of a weaker formulation of said problem $\endgroup$ – RGS Jun 6 '17 at 12:27
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    $\begingroup$ @RSerrao Thank you so much for the recommendation. I already watched the video regarding the four point problem. $\endgroup$ – Ananyo Bhattacharya Jun 6 '17 at 12:31
  • $\begingroup$ @AnanyoBhattacharya From your name, I think you are from my state. If you have no idea of rigor, first read any good text book of Real analysis, then read Munkres- Topology, that would be enough for starting topology. $\endgroup$ – MAN-MADE Jun 6 '17 at 12:38
  • $\begingroup$ Your questions show, that you are also interested in other things. So perhaps it makes more sense to study mathematics from the beginning. $\endgroup$ – Dietrich Burde Jun 6 '17 at 12:42
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    $\begingroup$ Yes, first study the basics needed for topology, like calculus, metric spaces, geometry, set theory, analysis etc. Your current questions show what you are thinking about. $\endgroup$ – Dietrich Burde Jun 6 '17 at 12:44
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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 !

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  • $\begingroup$ Do I need to study vector calculus or tensor as a prerequisite for studying topology? $\endgroup$ – Ananyo Bhattacharya Jun 6 '17 at 12:34
  • $\begingroup$ You don't need tensor at all ! On the other hand, a bit of knowledge of multivariable calculus can be useful (but absolutely not necessary) for Lee. And I think Munkres does not ask any prerequisites, so maybe you should begin by this (but I think Lee's book is perfect, so maybe give it a try :) ) Another argument in favor of Lee : its author is on this website and sometimes answering to questions :) $\endgroup$ – user171326 Jun 6 '17 at 12:37
  • $\begingroup$ My only complaint against Lee's book is that I didn't find it general enough, but it's a very good (really) introductory book. $\endgroup$ – YoTengoUnLCD Jun 6 '17 at 13:30
  • $\begingroup$ @YoTengoUnLCD : sure but it would be silly to give a very formal book for study topology first (e.g Bourbaki), I think it is the good balance. $\endgroup$ – user171326 Jun 6 '17 at 14:30
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Basic Topology by Armstrong. For me it was an awesome experience to read it. very inspiring!

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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.

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