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).

Are you sure you read correctly? Suppose $C$ is not connected in $X$. By definition this means that there exists two open sets $U, V\subset X$ such that $U\cap C \neq \emptyset$, $V\cap C\neq \emptyset$, $U\cap V = \emptyset$, and $C\subset U\cup V$. But then by definition of subspace topology, $U' = U\cap Y$ is open in $Y$, and $V' = V\cap Y$ is open in $Y$. And since $C\subset Y$, you have that $U' \cap V' = \emptyset$ and $C \subset U'\cup V'$, and $C \subset C\cap U \subset Y\cap U = U'$ etc. so $C$ cannot be connected in $Y$. 
