We have two spaces $X=\{(x,1/n):n\neq 0, n\in\mathbb{Z}, x\in\mathbb{R}\}$ and $Y=\{(x,n):n\neq 0, n\in\mathbb{Z}, x\in\mathbb{R}\}$. On both spaces we introduce the equivalent relation $(x,y)\sim (x',y')$ if $x=x'$ and $y=y'$ or $x=x'=0$. That is, all points on the $y$ axis are collapsed to the same point.
We are asked whether $X/\sim$ and $Y/\sim$ are homeomorphic in quotient topologies.
It is easy to show that the original spaces are homeomorphic. However, I don't know how to answer the question about the quotient spaces.
My guess is that they might not be homeomorphic and some problem might occur at the $origin$ but I am not sure.
Any hint would be helpful! Thanks!