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I was recently look at a post on tex.stackexchange about explaining $\LaTeX$ to the OP's grandmother. I was wondering, could the same thing be done for topology? Except in this case the "grandmother" is me. I have not fully understood the gist of topology and its capabilities. To my understanding, topology is the study of spaces but how does that translate into equations and variables? Anything would be helpful.

Thanks

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@GitGud I sure hope so, as the OP is 17. –  Alex Becker Jan 26 '13 at 19:01
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Are you a real world version of Philip J. Fry? –  Asaf Karagila Jan 26 '13 at 19:01
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Coming soon: Can I explain Mochizuki's approach to abc to my grandmother? –  Matemáticos Chibchas Jan 26 '13 at 19:01
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@GitGud I meant that I knew just as little as the OP's grandmother when it came to Latex or in this case topology. Therefore, I'm the "grandmother". –  gekkostate Jan 26 '13 at 19:03
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To emphasize the beginning of Asaf Karagila's answer, when you ask "how does that translate into equations and variables?" I infer that you have a far too restricted idea of what mathematics is. Vast parts of mathematics are about things totally different from equations and variables. –  Andreas Blass Jan 26 '13 at 19:50

4 Answers 4

up vote 10 down vote accepted

Topology, aka "rubber-sheet geometry", is when a teacup is identical to a donut but there is no way a teacup could ever be like this.

Topologists worry a lot about odd rings and bottles, some of them are quite concerned by knots while others try to comb hairy balls. All in all, these are rather strange characters...

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If my spoon has a little hole at the top of the handle? :-) Similar to Mrs. Pepper Pot (but without the bell). –  Asaf Karagila Jan 26 '13 at 19:34
    
@Asaf Hope you like the new example. –  Did Jan 26 '13 at 19:47

First you need to understand that mathematics not about equations and variables. It's about logical consequences from assumptions and definitions.

After understanding this, it would be wise to consider the fact that mathematics strives to abstract notions. We begin with a concrete object, say the real numbers, and we investigate its properties for a while. Then we realize that some of these can be transferred to a much broader generality. For example the idea of convergence, and the idea of "nearness". These translate to open sets, and general spaces.

Then we can ask, after we have the idea about what is an open set - what more can we say? And it turns out that we can say a lot. We can ask questions internal to the space itself:

  • Can we separate any two points by disjoint open sets?
  • Is there a countable set whose elements are "arbitrarily close" to any given point in space?
  • If we cover the space with open sets, can we find a finite subset of this cover which already covers the entire space?

Or we can ask questions related to the relation of this space to other space:

  • What sort of continuous functions are there from $[0,1]$ into our space?
  • Is there a structure which is compatible with our notion of "open sets" somehow?

There are many other directions to topology, in which I am not sufficiently familiar to write much, but this is likely to be remedied by other skilled users of this site.

All these things are very abstract already, but later can be realized to solve a concrete problem like how to build a bridge, or how to store data on your hard drive. This realization is far from a trivial process and often mathematicians don't see (and usually don't care) about such applications of the abstractness to the real world, and to the variables and equations.


Also interesting:

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Just be careful not to separate things too much. By the time Hausdorff separation comes along you can't model computer logic, according to Bishop. –  alancalvitti Jan 26 '13 at 20:00
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dear alancalvitti, where does Bishop actually say that? –  Jimmy R Jan 26 '13 at 20:29

I would recommend you read Chapter 10 Topology in Ian Stewart's book Concepts of Modern Mathematics. There the author explains, in very comprehensive language, the main ideas of topology and illustrates his discussion with many figures.

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Firstly, a topology is nothing but a subset of the power set of some set. This special set is required to satisfy some conditions(conditions that we intuit from some "concrete" objects, mainly Euclidean spaces) by definition. So this way, a topology, as a "compact"(do not confuse with term "compact space", I used its daily meaning) set, carries lots of information about the whole big set(which is your space). You can ask what kind of information we can get out of it? That is the crutial point indeed! Since a topology is a set of sets, it tells you in which way your points are connected/interrelated. For instance, due to its topology, some part of your space may perform wild behaviour while some other parts are kind of tame etc. Moreover, you can read from the topology where "too many" points are accumulated and where the less are at and so on...

A few words on "doughnot = coffee mug" issue which is always given as a cliche example to a beginner of topology... You consider, as a topologist, a doughnot and a coffee mug as the same; the same in the sense that you care only in which way the points of both shapes are related. Here, for example, you forget about "distance" matter(the distance between any two points) as we have stepped up in a generalization(abstraction) level.

I think the key point is, to mention again, a topology is a core which one can reach the information of how the points are "related".

(Not an organized entry but I would be appreciated if I could contribute a bit.)

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"Firstly, a topology is nothing but a subset of the power set of some set." My grandmother would have already quit listening. –  Austin Mohr Jan 26 '13 at 22:10
    
In case the grandmother is disturbed by so many "set"s, then let us just say, you collect some elements of the power set. –  Metin Y. Jan 26 '13 at 23:23

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