# How to understand this example in Do Carmo?

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?

## 1 Answer

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.

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