Let us consider
- $X=[-1,1]$
- $Y= \{(x,y)\in \mathbb{R}^2 | x=0, 0\leq y \leq 1/2 \} \cup \{ (x,y)\in \mathbb{R}^2 | y=0 , 0\leq x \leq 1/2 \}$
I hope they are not diffeomorphic as Y have a sharp corner. I don't know how to say this precisely.
Kindly help me with this