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).