Let $X=\mathbb{S}^{1}$ denote the unit circle and let:
$Y=\{(0,y) \in \mathbb{R}^{2}: -1 \leq y \leq 1\} \cup \{(x,0): 0 \leq x \leq 1\}$.
Prove that $X$ cannot be embedded in $Y$ and $Y$ cannot be embedded in $X$.
Well certainly I can see that $X$ and $Y$ are not homeomorphic, remove the origin $(0,0)$ from $Y$ then $Y \setminus \{(0,0)\}$ is not path connected while $X$ minus a point is. However I don't see how to prove $X$ cannot be homeomorphic to any subspace of $Y$ and vicerversa. Any ideas?