Topological spaces

A topological space is like a magical world where we can talk about how near or far things are from each other without relying on things like distance or measurement. Imagine you have a bunch of toys and you want to put them in different boxes based on how close they are to each other. You can do this by putting all the toys that are very close together in one box and all the toys that are not so close together in another box.

But how do you know which toys are "very close" and which are "not so close"? Well, you can use a set of rules that tells you which toys are allowed to be near each other based on their properties, like their shape or color. These rules are like the magic spell that creates your topological space.

Now, you can imagine that your boxes are like different parts of your magical world. Each box represents a region where the rules of the magic spell apply. So, toys that are close together in one box may not be close together in another box.

Topological spaces can be used to study many different things like the shape of the universe or the behavior of waves. The rules of the magic spell can be adjusted to suit the particular problem we are trying to solve. But always remember, a topological space is like a magical world where anything is possible.