Open and closed maps

Imagine you have a big box full of toys. Let's say you want to organize them in a special way. You decide to put all the red toys in one pile and all the blue toys in another.

An open map is like taking a picture of the box and being able to see all the different colors of the toys. You can clearly see which toys are red and which toys are blue. It's easy to tell how everything is organized.

On the other hand, a closed map is like looking at the box with a piece of paper covering it. You can still see some of the colors of the toys, but you can't see all of them. It's harder to tell how everything is organized, and you might miss some important details.

In math terms, an open map is a function that takes a set of points and maps them to another set of points in a way that preserves the structure of the original set. This means that if two points are close together in the original set, they will still be close together in the new set.

A closed map, on the other hand, takes a set of points and maps them to another set of points, but doesn't necessarily preserve the structure of the original set. This means that points that were originally close together could end up far apart in the new set, making it harder to analyze the structure of the original set.

In summary, open and closed maps are like looking at a box of toys with or without a piece of paper covering it. Open maps make it easy to see how things are organized, while closed maps make it harder to analyze the structure of the original set.