Explaining Topology

(Revised from a prior commentary):

You know, every time someone gives an elementary description of Topology (a branch of modern mathematics), there's a very standard explanation of it, and I think it's a very, very bad one. They always say something complicated like this (from http://www.sciencenews.org/articles/20071222/bob11.asp ):

Topology studies shapes. Specifically, it studies shapes' properties that are not affected by stretching, moving, twisting, or pulling—anything that doesn't break up the object or fuse some of its parts. The proverbial example is that, to a topologist, a coffee mug is the same as a doughnut. In your imagination, you can squash the mug into a doughnut shape, and it will retain the property of having a hole, namely its handle. A sphere is different. You can stretch a sphere into a stick and bend the stick so its ends touch. But turning that open ring into a doughnut will involve fusing the ends, and that's forbidden.

Huh? What the hell does that mean? You start off saying it's about shapes, then start talking in the negative by saying it's not about a bunch of particular properties of shapes. Then there are two pretty poor examples (asking people to imagine stretching things where bulky parts become very thin pieces; it's unclear what corresponds to what). I've taken a full year in graduate Topology, and sometimes I still have trouble understanding that description. Worst of all is this -- that's not what is really important about Topology studies. No one is ever really interested in stretching anything in a topology course.

Here's what I say in the classes I teach: Topology is the study of connections. That's the real story; it's very simple. Yes, coffee cups and donuts are similar topologically, because they're both connected bodies with one hole through each of them. But topology is really useful for things like the following -- A road engineer categorizes intersections by how many streets meet there. A miniature figure modeller plans how complicated an item they can sculpt, knowing the resulting mold has to stay connected around their figure. A stencil-maker has to make stencils one way for letters that have holes in them, and another way for those that don't (e.g., cut out an "A", "B", or "D" normally from paper and those middle holes get disconnected and fall out; that's not a problem for letters like "C", "E", or "F", which keep the surrounding paper connected.) A subway-rider looks for the easiest route to an evening out on the town, knowing they're restricted to specific connecting trains at specific stations. A traveling salesman wants to plan the fastest, cheapest sales trip between a dozen cities, using available commercial connecting flights; or, my food delivery service wants to do the same thing with intersecting city streets.

These are all Topological problems, dealing with how things are connected (which might be solid shapes, but is even more likely to be cords, knots, network circuits, or car/plane/train paths). I suspect I know why most explainers use the big-complicated-useless explanation, instead of the short-simple-and-effective one -- when categorizing different shapes, mathematicians do utilize functions called "homeomorphisms", which somebody at some point thought was best visualized as "stretching" operations. But, seriously, nobody who's nontechnical is going to care about that technique (no more than say, people care about how completing-the-square is used to develop the quadratic formula).

The point of all that technical work in Topology is, again, pretty simple: How is this shape connected? And hence: Where can I go today with this shape? That should be the focus of our first introductions to Topology, I think, not the damn "stretching" analogy, which is practically a cancer on our attempts to explain the subject.