Take a ball of soft clay. You may squeeze it, stretch it, dent it, roll it — anything, as long as you never tear it and never poke a hole through it. Everything you can reach this way counts, to a topologist, as the same shape. A cube, a pear, a pancake: all one thing. Topology is the study of what survives that kind of abuse. Size doesn't survive. Angles don't survive. Curvature doesn't survive. Almost nothing survives — and the few things that do are therefore very special: they are properties of the shape itself, not of any particular pose it happens to be in.
The most famous survivor is a hole. You cannot stretch a ball into a donut without tearing, and you cannot stretch a donut into a ball without sealing the hole shut. Here is the argument, and it needs no algebra: ask what a loop of string can do. Lay a loop anywhere on the surface of a ball. Whatever you do, you can always slide it along the surface, shrinking as it goes, until it collapses to a single point. Nothing stops it. Now thread a loop around the tube of a donut. Slide it however you like — it can travel all the way around the ring, it can wobble — but it can never shrink to a point, because the tube is always in the way. To free the loop you would have to cut it or tear the surface. No amount of smooth stretching changes which of these two situations you are in. That yes-or-no fact — can every loop shrink, or is some loop stuck? — is a property of the space itself.
“A topologist is someone who can't tell a coffee mug from a donut.” The joke is a theorem, and you can perform it. A mug has exactly one hole — the one through the handle. The bowl you drink from is not a hole: it's a dent, and by clay rules dents are free. So fatten the donut's ring on one side, press a hollow into the fat part, and you have a mug — no tearing, no gluing, at no point did the hole through the handle ever close or a new one open. Drive it yourself:
For closed clay shapes, the hole count is not just a label — it is essentially the label. Two blobs of clay can be stretched into each other exactly when their hole counts match (topologists call the count the genus, but "number of holes" is the whole content). Sphere: zero. Donut and mug: one. Eyeglasses frame: two. Pretzel: three. Everything else about the blob — lumpy or smooth, giant or tiny, elegant or hideous — is pose, not shape. Which means you can classify shapes the way topology does, right now, by playing:
Next: the word for spaces that look flat up close no matter how they curve as a whole — the single most-used unexplained word on this site.