What mathematical needs one needs to have to be willing to study topology?

Question

Suppose you enroll in a mathematics course with a special property: As you feel a certain important mathematical need which is identified by a professor, then you have to take lectures at that.

For example, in calculus, for most basic functions you can use a hand-waving approach for $h\to a$ and still get the right answer by substituting $h$ for $a$ at a certain point of the computation, they give certain trivial problems to show that sometimes the limits are not what they seem to be, such as limits in the form:

$$\lim_{x\to a}\frac{g(x)}{f(x)}$$

In which $f,g$ are polynomials and $f$ can be divided by $g$. But then, there are a lot of ways to construct functions, for example: Ways that use infinite processes, and for a lot of these examples, the handwaving approach fails. This shows that we can't just switch $h$ for $a$ at a certain point of the computation and that we need something better for computing these limits. If you perceived that, then you will need analysis lectures.

Other examples may need not only a better theory that explains a certain mathematical behavior$[1]$, but a theory that makes things easy, as Spivak says in his Calculus in Manifolds: "Chains and partitions of unity reduce questions about manifolds, where everything seems hard, to questions about Euclidean space, where everything is easy.

So in the case of topology, what mathematical phenomena are we trying to explain better? Or what are we trying to make easier?

For example, I am slightly aware that it has (also) to do with continuity but what are the elementary mathematical examples in which we need a better/truer/more economic theory (as the previously mentioned functions arising from infinite processes in the context of calculus/analysis)?

$[1]:$ In the previous example, not only a theory that better explains a mathematical behavior, but a theory that truly explains a mathematical behavior.

I am not sure what your question is, but if you look to the right of your question you see a couple of questions about motivation for topology, http://math.stackexchange.com/questions/456309/why-are-topological-spaces-interesting-to-study?rq=1, http://math.stackexchange.com/questions/207790/what-is-the-role-of-topology-in-mathematics?rq=1 — Moishe Kohan, Dec 05 '16 at 10:23
@MoisheCohen Yes, I have like 10 questions about it opened in my browser, but I feel that my question is slightly different of theirs: What I ask is not exactly the motivation, what I ask is - considering a background in, for example, calculus and linear algebra - what mathematical phenomena one needs to feel to be willing to study topology. — Red Banana, Dec 05 '16 at 17:15
@MoisheCohen A little bit of vector calculus. But very little complex analysis (for example, Apostol's Calculus V.1. The chapter on complex numbers and it's use to evaluate certain integrals). — Red Banana, Dec 05 '16 at 20:38
@MoisheCohen The problem I felt with the other answers on the motivation for topology is that they are too polished and it's a little bit difficult to use them to think about these elementary concepts developed at the subjects I mentioned. — Red Banana, Dec 05 '16 at 20:41
I see, then explaining topology would be very difficult. At the very least you need Green theorem , but better Gauss and Stokes theorems from vector calculus. There is a reason why topology got started in mid 19th century and not before: One reason is the needs of complex analysis and of vector calculus. My suggestion is to wait with your question until you are familiar with either one of these two or/and differential equations. — Moishe Kohan, Dec 05 '16 at 23:09
@MoisheCohen Do you think that the following introductions to topology are worthy readings? Frechet/Fan: Invitation to Combinatorial Topology; Blackett: Elementary Topology: A Combinatorial and Algebraic Approach; Henle: A Combinatorial Introduction to Topology; Boltyanskii/Efremovich: Intuitive Combinatorial Topology; Alexandrov: Elementary Concepts of Topology; Prasolov: Intuitive Topology; — Red Banana, Dec 07 '16 at 01:23
Any/all of those are good reads, but for something free you might try Robert Ghrist's Elementary Applied Topology. The "applied" business becomes more of a curiosity after chapter 5, IMO, but for the first few chapters it quite honestly talks about what sort of questions you might be thinking about to invite algebraic-topological concepts in your life. — Eric Stucky, Dec 08 '16 at 07:55
One way to answer the question is found in Chinn and Steenrod's First Concepts of Topology, which emphasizes the role topology plays in existence theorems for the solution of equations (algebraic and differential). This is one of the "needs" you ask about: when does an equation have a solution? One approach to the study of that question is topological. — symplectomorphic, Jun 27 '17 at 05:49

Lehs · Answer 1 · 2017-06-27T05:06:36.800

Point-set topology is an extension of set theory. In set theory the question is which elements that belongs to which sets and in point-set topology the question is which elements that are close to certain sets.

If $S$ is a set then $\bar S$ is the set of all elements that are close to $S$. This closeness can be defined in different ways and by different methods and one of them is described in

https://en.wikipedia.org/wiki/Kuratowski_closure_axioms

The mathematical phenomena explained by general topology is abstract closeness.

What mathematical needs one needs to have to be willing to study topology?

1 Answers1