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.

Let $\phi(x) = {1 \over 2x}$. Let $U = \{ (x,y) | x>0, \  y > \phi(x)\}$,
$V = \{ (x,y) | x>0, \  y < \phi(x)\}$. Then $U,V$ are disjoint, open, $S \cap U \neq \emptyset$, $S \cap V \neq \emptyset$ and $S \subset U \cup V$. Hence $S$ is not connected.
