Borel equivalence relation

So, imagine you have a bunch of toys, like blocks, dolls, and cars. You want to sort them into groups based on how they look. If two toys look exactly the same, they go in the same group. This is kind of like how a Borel equivalence relation works.

A Borel equivalence relation is a fancy math thing that helps us group things together based on certain properties they have. Let's say we have a bunch of numbers - some are even, some are odd. We could use a Borel equivalence relation to group the even numbers together and the odd numbers together.

But it's not just about sorting things based on one property. We could also use a Borel equivalence relation to group animals by their color, or cars by their brand. Basically, we choose a property to focus on and use that to group things together.

The important part is that the groups we create have to follow some rules. For example, if two things are in the same group, they have to be really, really similar - like two toys that look almost exactly the same. And if two things are in different groups, they have to be really different - like a doll and a block.

Overall, Borel equivalence relations help us organize things in a way that makes sense and follows rules.