Connected components of subspaces vs. space If $Y$ is a subspace of $X$, and $C$ is a connected component of $Y$, then C need not be a connected component of $X$ (take for instance two disjoint open discs in $\mathbb{R}^2$).
But I read that, under the same hypothesis, $C$ need not even be connected in $X$. Could you please provide me with an example, or point me towards one?
Thank you.
SOURCE http://www.filedropper.com/manifolds2 Page 129, paragraph following formula (A.7.16).

Isn't it just false? The image of a connected subspace by the injection $Y\longrightarrow X$ is connected...
