Consider $\mathbb{R}^2$ with usual topology. Let $S =\{(x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Z} \}$ . Then the set $S$ is ?
1) open but not closed
2) both open and closed
3) neither open nor closed
4) closed but not open ??
My attempt: I can able to see $S$ is closed set in $\mathbb{R^2}$. Since $\mathbb{R^2}-S$ is union of vertical strips which are open and hence $\mathbb{R^2}-S$ is open, so that $S$ is closed. But in key, it was given that it is open too, but I am not able to see, how it is open in $\mathbb{R^2}$. Please help me...