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 be a covering map, let .

Any path beginning at has a unique lifting to a path in E beginning at

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[My question relates to the proof of the uniqueness of ]

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 , such that for each i the set lies in an open set U (Use Lebesgue number lemma).

We define the lifting step by step.

First define .

Then supposing is defined for we define ] on as follows:

The set lies in some open set U that is evenly covered by p.

Let be a partition of into slices; each is mapped homeomorphically onto U by p.

Now lies in one of these sets. ... ... etc etc ... see attachement

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But now focussing on the uniqueness of - couldn't we define as belonging to any of the sets 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 for each of the - but this is not may idea of a unique .

Can someone please help clarify Munkres argument regarding the uniqueness of

Peter