I am interested in topology but I don't know if I can learn it without learning something else first.
I've done: Algebra 1 and 2 Euclidean Geometry Calculus
Is that enough if not please tell me what else I need to learn.
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asked Dec 11, 2014 at 19:08
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3$\begingroup$ Ask your teacher, no general answer can be given. $\endgroup$– Michael GreineckerCommented Dec 11, 2014 at 19:10
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3$\begingroup$ For point-set topology, there are almost no pre-reqs, other than the ability to reason from definitions. Algebraic topology requires abstract algebra (it's not clear what you mean by Algebra, but it sounds like you are using the high school term, which is different from abstract algebra.) $\endgroup$– Thomas AndrewsCommented Dec 11, 2014 at 19:11
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2$\begingroup$ The only thing I would suggest is to have had a course which emphasizes formal proof. This is typically what is called "Advanced Calculus" or "Analysis", and is usually a upper-level undergraduate course. Don't let the word "calculus" in the title mislead you -- it isn't the calculus you typically see in high school or undergraduate calculus courses; it's focus is "grown-up" rigorous mathematical techniques. If you don't know them, you will have a hard time in a topology course. That said, it of course depends on the level of difficulty of the particular course you are considering. $\endgroup$– MPWCommented Dec 11, 2014 at 19:18
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4$\begingroup$ I'd say you should have some understanding of how mathematical proofs are written and some acquaintence with naive set theory. $\endgroup$– Michael HardyCommented Dec 11, 2014 at 19:24
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2 Answers
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Set theory (naive set theory is fine for the most part, axiomatic set theory can sometimes be relevant) and a good grounding in reading and writing mathematical proofs are the two essentials for point-set topology.
Anything else you know won't be strictly necessary, but it will put definitions and examples in the proper context. Some knowledge of calculus or real analysis gives you a feel for the abstract definitions of convergence and continuity. If you know some group theory you will be able to talk about topological groups and orbit spaces, which gives you more examples of topological spaces to think about. You will also be able to get into algebraic topology later on.
Topology is a subject where understanding basically rests on the number of examples you have handy, and also a very widely encompassing subject. So with more background in other subjects you will have an easier time with obtaining a conceptual understanding.
answered Dec 11, 2014 at 19:21
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You should know basic operations in set theory.
For example, do you think that:
$f(A\cup B)=f(A)\cup f(B)\: ? \: f(A\cap B)=f(A)\cap f(B)\: ? \:f(f^{-1}(S))=S\:? \: f^{-1}(f(T))=T \:?$
...and can you state a precise context in which these questions make sense?
answered Dec 11, 2014 at 19:28