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I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example in which I cannot understand the explanation he gave. enter image description here

enter image description here

enter image description here

I really don't understand what he said about why $\alpha$ is not an embedding... No worry about my knowledge on topology. Can anyone help me “translate” it to the common language that's easy to understand?

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The set $C=\alpha\bigl((-3,0)\bigr)$ has two topologies:

  • the topology it inherits from the usual topology in $\mathbb{R}^2$: a set $A\subset C$ is open if there is an open subset $A^\star$ of $\mathbb{R}^2$ such that $A=A^\star\cap C$.
  • the topology it gets from $(-3,0)$: a set $A\subset C$ is open if there is an open subcet $A^\star$ of $(-3,0)$ such that $A=\alpha(A^\star)$.

Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is not.

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    $\begingroup$ Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism? $\endgroup$ – user450201 Nov 12 '18 at 11:27
  • $\begingroup$ That would be correct. On the other hand, that is what Do Carmo is claiming. $\endgroup$ – José Carlos Santos Nov 12 '18 at 11:34

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