Sequentially compact space

Alright kiddo, let's talk about sequentially compact space.

Imagine you have a really big toy box with lots of toys inside. You want to pick out all the toys one by one, but you can only pick out a certain number of toys at once. Let's say you can only pick out three toys at a time.

Now, imagine that there are so many toys in the box that you could pick out toys forever and still never finish picking them all out. But, let's say that no matter how long you keep picking out toys, you always end up picking out three toys at a time.

In math terms, we call this idea "sequentially compact." It means that even though there are an infinite number of things in a set (like the toys in the toy box), we can still pick out subsets of that set (like the three toys at a time) so that these subsets have a limit point (like the number of toys you have picked out) that is still within the original set.

So, a sequentially compact space is just a fancy way of saying that we can keep picking out subsets of a set over and over again, and we'll always find a limit point that still belongs to the original set.

Hopefully that makes sense! If not, maybe we can play with some toys together and see if it helps.