Path

Let \(X\) be a topological space and \(I\coloneqq[0,1]\). A path in \(X\) is a continuous map \(\gamma\colon I\to X\); it is a path from \(\gamma(0)\) to \(\gamma(1)\). A path \(\gamma\) is a loop if \(\gamma(0) = \gamma(1)\), and then it is a loop based at \(\gamma(0)\). For paths \(\gamma, \eta\colon I\to X\) with \(\gamma(1) = \eta(0)\), the concatenation \(\gamma\cdot\eta\colon I\to X\) is the path \[ (\gamma\cdot\eta)(s)\coloneqq \begin{cases} \gamma(2s) & 0\leq s\leq 1/2, \\ \eta(2s-1) & 1/2\leq s\leq 1. \end{cases} \]