Betti's theorem

Okay, so imagine you have a big bowl of spaghetti with lots of loops and twists. Now, imagine you have a couple of strings and you want to see how many loops or holes you can wrap the strings around without them touching or crossing each other.

Betti's theorem is a fancy way of counting those loops or holes. It tells you that the number of loops or holes you can wrap your strings around is equal to the number of closed loops minus the number of strings that touch those loops plus one.

Let's say you have two strings, and you can wrap them around two separate loops without them crossing each other or touching the loops. Then, according to Betti's theorem, you would have two closed loops and no strings touching, which gives you a total of two. Then add one, and you have three. So there are three loops or holes that you can wrap your strings around in that big bowl of spaghetti!

In math terms, Betti's theorem is a way to calculate the topological invariants - or features that don't change when you bend or twist an object - of a certain type of object called a simplicial complex. This can include things like graphs, networks, and more complex shapes. It's a useful tool for studying the shape and structure of objects in different fields, including mathematics, physics, and engineering.