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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?

Ananyo Bhattacharya
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  • 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 – RGS Jun 06 '17 at 12:27
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    @RSerrao Thank you so much for the recommendation. I already watched the video regarding the four point problem. – Ananyo Bhattacharya Jun 06 '17 at 12:31
  • @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. – MAN-MADE Jun 06 '17 at 12:38
  • Your questions show, that you are also interested in other things. So perhaps it makes more sense to study mathematics from the beginning. – Dietrich Burde Jun 06 '17 at 12:42
  • @DietrichBurde Sorry but I didn't get your point. Study maths from beginning? – Ananyo Bhattacharya Jun 06 '17 at 12:43
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    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. – Dietrich Burde Jun 06 '17 at 12:44
  • @DietrichBurde In geometry will it be enough to study Euclidean geometry ? – Ananyo Bhattacharya Jun 06 '17 at 12:50
  • This depends on the meaning of "enough" for you. To have more understanding and fun for doing mathematics, I think you should not try to minimize prerequisites, but rather do what you find interesting immediately, e.g., why not hyperbolic geometry. – Dietrich Burde Jun 06 '17 at 12:57
  • There are different areas referred to as topology. If you have in mind areas like algebraic or differential topology, which is what the term often refers to in math writing aimed at the general public, then these areas are too advanced for you at the moment. What might be accessible now would be basic facts about metric spaces. For example, you could read the beginning of Apostol's Mathematical Analysis, or have a look at Kaplansky's Set Theory and Metric Spaces. However, you might also find this material too difficult. In that case, you should wait until later to study topology. – user49640 Jun 06 '17 at 21:43
  • Topology is an abstraction of open sets and their continuous transormations in Euclidean space, so you really should beging with calculus and mathematical analysis. Analysis also shows you the true value of general, algebraic or differential topology. – Hulkster Jun 06 '17 at 23:24
  • There are many, many roads to learn topology. I am more or less convinced that the easiest and the most motivating introduction is calculus -> multivariable calculus -> differential topology (even though this is not the study-path I used). – Balarka Sen Jun 07 '17 at 10:45

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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 !

RGS
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  • Do I need to study vector calculus or tensor as a prerequisite for studying topology? – Ananyo Bhattacharya Jun 06 '17 at 12:34
  • 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 :) –  Jun 06 '17 at 12:37
  • 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. – YoTengoUnLCD Jun 06 '17 at 13:30
  • @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. –  Jun 06 '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.

Thomas Andrews
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