Consider the set \( x {0})

Can I say that the sets

U:= {(x,y) in \( x {0}):y>0}

V:= {(x,y) in \( x {0}):y<0}

form a separation of \( x {0}), meaning that it is not (path) connected?

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