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I am reading Munkres book Topology.

Currently, I am studying Section 54: The Fundamental Group of the Circle and need help with a minor point in the proof of Theorem 54.4

Theorem 54.4 and its proof reads as follows:

In the proof we read:"If [itex]E[/itex] is path connected, then, given [itex]e_1 \in p^{-1}(b_0)[/itex], ... ... "... ... BUT ... ... how do we know there exists an [itex]e_1[/itex] different from [itex]e_0[/itex] in [itex]p^{-1}(b_0)[/itex] ... maybe [itex]e_0[/itex] is the only element in [itex]p^{-1}(b_0)[/itex]?

What, indeed, do we know about the nature of [itex]p^{-1}(b_0)[/itex]?

Hope someone can help ...

Peter

Currently, I am studying Section 54: The Fundamental Group of the Circle and need help with a minor point in the proof of Theorem 54.4

Theorem 54.4 and its proof reads as follows:

In the proof we read:"If [itex]E[/itex] is path connected, then, given [itex]e_1 \in p^{-1}(b_0)[/itex], ... ... "... ... BUT ... ... how do we know there exists an [itex]e_1[/itex] different from [itex]e_0[/itex] in [itex]p^{-1}(b_0)[/itex] ... maybe [itex]e_0[/itex] is the only element in [itex]p^{-1}(b_0)[/itex]?

What, indeed, do we know about the nature of [itex]p^{-1}(b_0)[/itex]?

Hope someone can help ...

Peter