Homotopy class

Imagine that you have to go from one place to another on a big sheet of paper. You can draw a line on the paper which shows your path. Now imagine that there are many different ways to draw that line from the start to the finish point. These different ways of drawing the line are called homotopy classes.

To understand it better, draw two points on the paper and try to connect them with a line. You can make a straight line or a curved line with many different curves - each with different shapes. Each of these different lines represents a different homotopy class.

A homotopy class is like a group of all the different ways you can draw a line from one point to another point without lifting your pencil off the paper. These different lines may look different, but they belong to the same class because they can be transformed from one to the other just by stretching and moving them around without crossing over any other lines on the paper.

So, basically, a homotopy class is a collection of all the different paths you can take between two points on a surface, all of which can be transformed into each other without lifting your pencil off the surface.