Gluing axiom

The gluing axiom is kind of like when you put together a puzzle. Imagine you have a really big puzzle, but you don't have enough space to put it all together at once. One solution is to break the puzzle up into smaller pieces, put them together separately, and then glue them all together to make the big puzzle.

In math, sometimes we have things called manifolds, which are kind of like shapes or spaces that we can study. A manifold might be curved, twisted, or have all sorts of weird shapes, but we can still use math to understand it.

Sometimes, though, we can't study the whole manifold at once, just like we can't put together a big puzzle all at once. The gluing axiom comes in handy in situations like this. It helps us study parts of a manifold separately, and then glue them together to get a better understanding of the whole thing.

Just like with the puzzle, we have to make sure that the pieces we're gluing together fit perfectly. We don't want any gaps or overlaps, because that would mess up our understanding of the whole manifold.

So, the gluing axiom is a way for mathematicians to take big, complicated shapes and spaces, break them up into smaller pieces, study those pieces separately, and then glue them back together to understand the whole thing.