In which of the following two cases, $X$ is homeomorphic to $Y$?
$X = \{(x, y, z)\in \mathbb{R}^3: x^2+y^2=1\}, \ Y = \{(x, y, z)\in \mathbb{R}^3: z=0, x^2+y^2\neq 0\}$
$X = \{(x, y, z)\in \mathbb{R}^3: x^2+y^2=1\}, \ Y = \{(x, y, z)\in \mathbb{R}^3: x^2+y^2=z^2\neq 0\}$
I am able to get the $X$ is an infinite cylinder in both the options. $Y$ in the first option is complete xy plane except $(0, 0, 0)$. So, how to look further?
In the case of the second option, $X$ and $Y$ are not homeomorphic, since $X$ is connected, whereas $Y$ isn't.
On the other hand, in the case of the first option, we do have a homeomorphism:$$\begin{array}{ccc}X&\longrightarrow&Y\\(x,y,z)&\mapsto&e^z(x,y,0).\end{array}$$