Kuratowski's theorem

Imagine you have a really big piece of paper and you want to draw a picture on it, but there are some rules you have to follow. Kuratowski's theorem is like a set of rules for the kind of picture you can draw on that paper.

Kuratowski's theorem says that you can't draw a certain kind of picture on that piece of paper. This picture has two points (let's call them A and B) that are not connected by a line, but every other point on the paper is connected to either A or B (or both) by a line. It's like if A and B were on opposite sides of the paper and everything else was a spider web of lines connecting them.

So why can't you draw this picture? Well, imagine that the paper is actually a tablecloth and you want to fold it in half so that A and B touch each other. If you try to do that with the spider web picture, the lines will get all crumpled and tangled up and you won't be able to fold it neatly.

Kuratowski's theorem basically says that if you can draw a spider web picture like this, then it's impossible to fold the paper neatly. This is really important in math because it helps us understand how different shapes are related to each other and what kinds of shapes we can create without breaking the rules.