Finitely generated space

Imagine you have a big toy box filled with all kinds of toys. Some toys can be made by putting different pieces together, like a puzzle. Others, like a ball or a teddy bear, are already whole and can't be taken apart.

A finitely generated space is like a toy box. But instead of toys, we have points and lines. Some points can be made by putting different lines together, like building blocks. Other points are already there and can't be broken into smaller pieces.

So in a finitely generated space, we have a bunch of points and lines that we can use to build other points and lines. But we can't keep building forever - there's only a limited number of building blocks we have to work with. No matter how big our space is, we can always describe it with a finite number of building blocks.