I am a graduate student.In this semester we have a topology course with a basic introduction to fundamental groups and covering spaces.While studying covering spaces,I encountered the following theorem which is called the homotopy-lifting lemma.It is as follows:
Let $F:I\times I\to X$ be a continuous map such that $F(0,t)=F(1,t)=x_0$ for $t\in [0,1]$.Then there is a unique continuous map $\tilde F:I\times I\to \tilde X$ such that $p\circ \tilde F=F$ and $\tilde F(0,t)=\tilde x_0,t\in [0,1]$.
Where $p:\tilde X\to X$ is a covering map and $I=[0,1]$ and $\tilde x_0\in p^{-1}(x_0)$.
I want a proof of this theorem and also some intuition of what is going on.Can someone help me?