Which two of the following space are homeomorphic to each other?

\begin{align}
X\_1&=\{(x,y)\in \mathbb{R}^2:xy=0\}\\
X\_2&=\{(x,y)\in \mathbb{R}^2:xy=1\}\\
X\_3&=\{(x,y)\in \mathbb{R}^2: x+y\geq 0 \text{ and } xy=0\}\\
X\_4&=\{(x,y)\in \mathbb{R}^2: x+y\geq 0\ \text{ and } xy=1\}
\end{align}

I want to use the contentedness property.

Efforts: 

$X\_1$ and $X\_4$ are not homeomorphic as $X\_1$ is connected and nd $X\_2$ is disconnected(two connected component)

By same logic $X\_1$ and $X\_4$ are not homeomorphic.

If I remove the origin from $X\_1$, there are 4 components, but if we remove the origin from $X\_3$, there are three connected component. So $X\_1$ and $X\_3$ are not homeomorphic. 

Am I going in the right direction?

How to proceed after this.