quality:corruption
Definition

Throughout this article, X is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, x 0 {\displaystyle x_{0}} is a point in X called the base-point. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on X can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.