How to understand this example in Do Carmo?

Question

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

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?

Answer 1

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.