Kdjykuj

2 $begingroup$ 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+ygeq 0 text{ and } xy=0}\ X_4&={(x,y)in mathbb{R}^2: x+ygeq 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.