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

Solution 1:

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...

Solution 2:

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:

• The definition of metric space,topological space