How can I explain topology to my grandmother? 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
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...
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.)