Consider the set $\displaystyle \mathbb{R}^2$\($\displaystyle \mathbb{R}$ x {0})

Can I say that the sets

U:= {(x,y) in $\displaystyle \mathbb{R}^2 $\($\displaystyle \mathbb{R}$ x {0}):y>0}

V:= {(x,y) in $\displaystyle \mathbb{R}^2 $\($\displaystyle \mathbb{R}$ x {0}):y<0}

form a separation of $\displaystyle \mathbb{R}^2$\($\displaystyle \mathbb{R}$ x {0}), meaning that it is not (path) connected?

Also, I'm not sure whether the set is open, closed, clopen...any ideas?