prove that a space is not connected Let $S=\{(x,0)\} \cup\{(x,1/x):x>0\}$. Prove that $S$ is not a connected space (the topology on $S$ is the subspace topology)
My thoughts: Now in the first set $x$ is any real number, and I can't see that this set in open in $S$. I can't find a suitable intersection anyhow.

The set $\{(x,0) : x\in\mathbb R\}$ is open in $S$ because every point $(x,0)$ has an open neighborhood that does not intersect the graph of $y=1/x$.  Just use $I\times\{0\}$ where $I$ is any open interval containing $x$.
Then do a similar thing with the set $\{(x,1/x): x>0\}$.
