In Section 54 of his book "Topology" on the Fundamental Group of the Circle, Munkres presents the following Lemma (Lemma 54.1 - see attachment)

Lemma 54.1

Let $\displaystyle p: E \rightarrow B $ be a covering map, let $\displaystyle p( e_0 ) = b_0 $.

Any path $\displaystyle f : [0.1] \rightarrow B$ beginning at $\displaystyle b_0 $ has a unique lifting to a path $\displaystyle \tilde{f} $ in E beginning at $\displaystyle e_0$

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[My question relates to the proof of the uniqueness of $\displaystyle \tilde{f} $ ]

Munkres begins the proof as follows:

Proof:

Cover B by open sets U each of which is evenly covered by p.

Find a subdivision of [0.1], say $\displaystyle s_0, s_1, .... s_n $ , such that for each i the set $\displaystyle f([s_i, s_{i+1} ])$ lies in an open set U (Use Lebesgue number lemma).

We define the lifting $\displaystyle \tilde{f} $ step by step.

First define $\displaystyle \tilde{f} (0) = e_0 $.

Then supposing $\displaystyle \tilde{f} (s) $ is defined for $\displaystyle 0 \leq s \leq s_i $ we define $\displaystyle \tilde{f} $] on $\displaystyle [s_i , s_{i+1}$ as follows:

The set $\displaystyle f([s_i, s_{i+1} ])$ lies in some open set U that is evenly covered by p.

Let $\displaystyle \{ V_{\alpha} \} $ be a partition of $\displaystyle p^{-1} (U) $ into slices; each $\displaystyle \{ V_{\alpha} \} $ is mapped homeomorphically onto U by p.

Now $\displaystyle \tilde{f} (s_i) $ lies in one of these sets. ... ... etc etc ... see attachement

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But now focussing on the uniqueness of $\displaystyle \tilde{f} $ - couldn't we define $\displaystyle \tilde{f} (s) $ as belonging to any of the sets $\displaystyle \{ V_{\alpha} \} $ and make this work?

So any of the sets would do since the covering is even.

Then the argument for uniqueness would follow (see page 342 of attachment) - so we would only have one $\displaystyle \tilde{f} (s) $ for each of the $\displaystyle \{ V_{\alpha} \} $ - but this is not may idea of a unique $\displaystyle \tilde{f} $.

Can someone please help clarify Munkres argument regarding the uniqueness of $\displaystyle \tilde{f} $

Peter