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Starting to study general topology, one needs to invest time to go through many definitions and axioms. To get motivation, I would like to know what is a positive knowledge general topology will bring me. Could you briefly list a cornerstone theorems and tools which general topology will grant to me after studying. For instance, Calculus will teach me integration and therefore I will be able to find volume or area of any object; linear algebra teaches how to transform matrices and solve liner equations etc.

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    $\begingroup$ Just a remark (I am too poor in this to give an answer). Are you familiar yet with continuous functions and open sets? You will meet them again in topology in much broader and brighter perspective. $\endgroup$
    – drhab
    Commented Apr 17, 2017 at 8:24
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    $\begingroup$ Topology definitely revolves around openness and continuity. $\endgroup$
    – Arthur
    Commented Apr 17, 2017 at 8:32

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Personally the "useful tool" topology gives me is the ability to visualise almost anything in maths. For example, if you have some algebraic structure you may know how each element behaves in relation to the other elements, but it's hard to know what the overall structure "looks like". However, you can define a topology on it, and develop your own visualisation for the structure.

However, in terms of "useful theorems", you'll learn that a huge number of things that are true in real numbers can be generalised to general spaces with specific topological properties. For example, you'll learn about concepts such as "compactness" which give you generalisations of the extreme value theorem: "A continuous function defined on a compact set attains its maximum and minimum somewhere on the set".

Personally rather than gaining some "tool", the main reason I study topology is because the open problems in topology are so interesting. I'm doing research in continuum theory, specifically in an area called "generalised inverse limits". I've learned a lot about continua which can be embedded in $\mathbb{R}^2$, and it turns out that they're remarkably complex. Unfortunately I don't think you have quite enough understanding at the moment, but let me just say we're no where near classifying plane continua in any way. This is very surprising because before you study them you really don't expect them to be so complicated.

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