Let \(p\colon E\to B\) be a continuous map and \(I=[0,1]\). The map \(p\) has the path lifting property if for every path \(\gamma\colon I\to B\) and every \(e_0\in \inverseof{p}(\gamma(0))\) there is a path \(\tilde\gamma\colon I\to E\) with \(p\circ\tilde\gamma=\gamma\) and \(\tilde\gamma(0)=e_0\).
Identifying \(\pt\times I\) with \(I\), this is the homotopy lifting property of \(p\) with respect to the one-point space \(Y=\pt\).
Let \(p\colon\tilde X\to X\) be a covering space. For every path \(\gamma\colon I\to X\) and every \(\tilde x_0\in \inverseof{p}(\gamma(0))\) there is a unique path \(\tilde\gamma\colon I\to\tilde X\) with \(p\circ\tilde\gamma=\gamma\) and \(\tilde\gamma(0)=\tilde x_0\) [@Hat02].