Non-mathematician here. Topology sounds pretty cool. What do I need to learn to study topology on my own?
Have only had calculus.
Picture unrelated.
Basic set theory.
>>7688214
graph theory, algebra, and mathematical logic. after that you can get into topology, and eventually manifolds.
Introductory topology book.
>>7688243
you don't need more than five pages worth of material in either of those subjects, so don't worry.
On the other hand, topology can be pretty dry when you do not understand what to use it for.
That's why you usually learn it after analysis
>>7688249
Oh I see. Have you got any tips on where to start? I was skimming this Munkre book on the internet and it seems to have an introductory chapter with some set stuff. Would that be enough?
>>7688264
Not the same Anon. I'm studying pure mathematics at university now, but prior to that I was very interested in topology. Like others said, it doesn't require much knowledge: in fact, given a few notions of set theory and perhaps Analysis you are definitely able to grasp the ideas behind it. It's similar to algebra in that sense: you don't need anything in order to understand algebra but algebra itself. However, like algebra, topology can be a little confusing if you don't know where it can be applied and why it was born.
Following the algebra metaphor, one could easily learn what a ring is (really, it's just a sequence of definitions, much like almost all math), but first you must understand that algebraic structures rise from the way sets of numbers "behave". So while algebra helps understanding numbers with a theory that generalizes sets of numbers, topology helps you understand "spaces" with a notion that arises from the abstract concepts of closeness, compactness ecc. These are concepts used in (real) analysis and were first introduced in the calculus of variations.
So, yes you can definitely learn and understand topology, but I suggest you start from the basics and go further. I thought I understood topology, but the truth is that math is ALL connected, so in order to have a strong grasp or at least a correct intuition on a particular subject it is required to be at least knowledgeable in other fields. Munkres is difficult for a beginner, anyway. I suggest you start by learning the basics of abstract algebra and real analysis. After that you'll be prepared to understand at least point-set topology. General topology is even more abstract. The next level would be algebraic topology, but that is far from being an easy task.
You must have a lot of patience, dedication and most of all DO ALL THE EXERCISES and read the examples. Read every proof until you're sure you understand it, and try "reversing" definitions.
>>7688214
>What do I need to learn to study topology on my own?
You need to fuck off back to /pol/. Thats what you need to do.
Hope that helps
>>7688414
And you need to go back to /b/ where your useless comments are better appreciated. Thanks.
>>7688526
listen to this guy op, undergrads abound
>>7688526
Not necessarily true. I tried self-teaching topology from Munkres during my undergrad, but the book was a bit dry even if it was well-structured.
I went back to it after a course in analysis, and it was much more interesting and motivated.
>>7688264
Yeah, that definitely is sufficient
In case you don't understand something about logic, set theory or abstract algebra you can look it up on the spot and you should be fine.
>>7688555
>a copy of, say, Munkres
Munkres was not the point of my comment; I do not preclude the possibility that there are other good introductory books. I didn't learn topology from Munkres, personally.
>the book was a bit dry
>I went back to it ... and it was much more interesting and motivated.
OP doesn't need motivation for topology; he's not asking about learning mathematics in general, he's asking about learning topology specifically. If he finds topology's not as interesting as its pretty pictures, he can try to learn something else instead.
Also, anybody interested in learning some mathematical field (or many other things) should get used to dry writing as soon as possible. I'm not saying dry writing is good, but there are many times when the best source for something is far from a pleasure to read, and being able to deal with this is a crucial skill.
For somebody learning mathematics in general and wondering when to learn topology, I would agree with you that it's easier to swallow after a healthy dose of metric space topology from analysis.
You need a first course in analysis and algebra family.
>>7688214
Assuming you understand basic set theory, just pick up Munkres or Willard and start working.
For that matter, both of them have recap sections for it, so you could pretty much just go straight to them.
>>7688657
Fair enough, but OP shouldn't specifically digest uninteresting material just to "get used to it." Unless he's planning on going into research, he can pretty much always avoid it.
Some set theory and then Mendelson for dummies with no grasp.
>>7688407
>>7688526
>>7688571
>>7688657
>>7689273
Useful comments, thanks guys.
The general consensus seems to be that I should learn a bit of algebra and analysis first, so I will look into that.
>>7688414
Don't worry, I dislike /pol/ as much as every other sensible person.
Doesn't mean I can't appreciate a pink Hitler though.
>>7689954
I love algebra, and I think you will get a lot out of learning the basics, but it's not really necessary for basic point-set topology. Things like quotient topologies can seem complicated without previous experiences with quotient objects in algebra, but it's not a killer. If you decide to move onto algebraic topology (this is the second half of the recommended Munkres), then you will need some topology, but really only the basics of groups.
But basic real analysis (make sure your book takes the metric space approach! I used Rosenlicht's book -- it's less than $10 on Amazon) will cause everything to at least feel motivated.
>>7689954
>The general consensus seems to be that I should learn a bit of algebra and analysis first
How did you deduce that from these comments?
>>7688214
Technically all you need is basic set theory, but getting through analysis and abstract algebra would be a great asset in familiarizing yourself with sets.
