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?
Yes, because we are dealing with strict inequality, the epsilon-neighborhood of every point in say, U is also in U (analogously for V).
Regarding open/close of in , am I correct in the following:
, being the union of open sets, is open in .
and because is in the boundary of but not in , it follows that is not closed?
I'm really not certain about the "NOT closed" part...