Let $p:E\rightarrow B$ be a covering map, let $p(e_0)=b_0$. Any path $f:[0,1]\rightarrow B$ beginning at $b_0$ has a unique lifting to a path $\bar{f}$ in $E$ beginning at $e_0$.
This is a lemma from the book "Topology" by Munkres. I am unable to understand first part of the proof.
Proof: Cover $B$ by by open sets $U$ each of which is evenly covered by $p$ . Find subdivision of $[0,1]$ , say $s_0,s_1,…,s_n$, such that for each $i$ the set $f([s_i,s_{i+1}])$ lies in such an open set . Using Lebesgue number lemma. How to use Lebesgue number lemma here??
Best Answer
Suppose you are given a covering of $B$ by open sets $\{U_\alpha\}_{\alpha \in A}$.
By applying continuity of $f : [0,1] \to B$, it follows that each set $V_\alpha = f^{-1}(U_\alpha)$ is an open subset of the topological space $[0,1]$.
By applying the fact that the open sets $\{U_\alpha\}_{\alpha \in A}$ are a covering of $B$, it follows that the open sets $\{V_\alpha\}_{\alpha \in A}$ are a covering of the topological space $[0,1]$.
There you are: You've got a compact space $[0,1]$, and an open covering $\{V_\alpha\}_{\alpha \in A}$, so you're ready to apply the Lebesgue Number Lemma. What it gives you is a number $\lambda > 0$ such that for every subset $S \subset [0,1]$, if the diameter of $S$ is $\le \lambda$ then there exists $\alpha \in A$ such that $S \subset V_\alpha$.
But now there's one more step: To apply the Lebesgue number $\lambda$ that we've found, we shall subdivide the open set $[0,1]$ into certain subintervals each of which has diameter $<\lambda$. To do this we choose an integer $n \ge 1$ such that $\frac{1}{n} < \lambda$, and then we subdivide using $s_i = \frac{i}{n}$. Each subinterval $[s_{i-1},s_i]$ has diameter $\frac{1}{n} < \lambda$. It follows that for each $i$ there exists $\alpha_i \in A$ such that $$[s_{i-1},s_i] \subset V_{\alpha_i} $$ from which you conclude that $$f[s_{i-1},s_i] \subset U_{\alpha_i} $$
By the way, you'll find that this method of applying the Lebesgue Number Lemma is used over and over and over again in Algebraic Topology, so this is a good place to master it: Using continuity of a certain function $f$ defined on a certain compact metric space, you pull back an open cover of the codomain of $f$ to get an open cover of the domain to which the Lebesgue number lemma is then applied.