fundamental group

The fundamental group is a concept in algebraic topology that helps classify geometric regions by studying paths within them. It is a set of paths defined on a space, where paths are considered equivalent if they can be continuously deformed into one another without leaving the space.123

Specifically, the fundamental group is formed from closed paths that start and end at the same point. If a region has no holes, all such paths are homotopic, and the fundamental group has only one element. In regions with holes, paths that wind around the hole a different number of times are not homotopic, leading to a more complex group structure. This algebraic structure provides a way to distinguish between different topological spaces based on their path connectivity.21