Carathéodory's criterion

Carathéodory's criterion is like a rule that helps us decide if an object is a point or a space in math. Imagine you have some things, like toys, and you want to put them together in a group. But you don't want to call that group just one thing if it really has lots of parts.

So Carathéodory's criterion says that if you have a toy box with lots of toys in it, you can only call it one toy if you can reach all of the toys by just using three moves. For example, you could move the toy box to a different place in the room, open the lid of the toy box, and take out any toy you want. If you can do all of that in just three moves, then you can call the whole toy box just one toy.

This rule helps us make math easier by grouping things in a way that makes sense. Just like putting toys into groups, we can put math objects into groups too. Carathéodory's criterion is an important tool that helps us decide when it's okay to group different math things together.