Compactly generated space

Okay, so imagine you have a really big room with a lot of stuff in it. And you have to clean up all that stuff, but you don't want to spend all day doing it.

So instead of picking up every single little thing, you decide to just pick up the big things that take up a lot of space. This makes it much easier and quicker to clean up the whole room, right?

Well, a compactly generated space is kind of like that. It's a type of space in math that has a lot of points, but instead of looking at every single point individually (which would take forever), we just look at the bigger parts of the space. These bigger parts are called "compact sets".

By just looking at the compact sets, we can still understand a lot about the whole space without having to examine every little point. And that makes things much easier for mathematicians who want to study these spaces.

So, to sum it up, a compactly generated space is a type of space in math that we can understand by just looking at the larger parts of the space, called compact sets. This allows us to study the space without having to look at every single point, saving us a lot of time and effort.