Homeomorphism group

Imagine you have lots of different kinds of toys. Some are flat, some are round, some are tall, some are short, some are very different from others. Now imagine you want to see how you could move these toys around to make them look exactly the same, like a reflection in a mirror. If you can do that, then those toys are said to be homeomorphic to each other.

Now imagine you have lots of different shapes, like circles, squares, triangles, and rectangles. You might be able to move them around and change their size or orientation to make them look like each other. If you can do that, then those shapes are also said to be homeomorphic to each other.

The homeomorphism group is like a big group of all the ways you can move things around so they look like each other. It's like having a giant toy box with all kinds of toys and shapes, and trying to see how you can move them around to make them look the same. It's a way of studying how shapes can be changed or transformed while still keeping their essential properties, like their size or number of corners. This can be used in many different fields, like math, physics, or even art and design!