Prove that the Complex plane is closed, open and perfect.
My intuition is destroyed by the fact that a set can be open and closed at the same time.
The following is my understanding.
open: If all points in set $E$ is interior to $E$, then $E$ is open.
I think this means that all points $p$ in $E$ has a neighborhood that is a proper subset of $E$.
closed: If every limit point of $E$ is a point of $E$, then $E$ is closed.
I think this means that all neighborhoods of every limit point $p$ in $E$ contains a distinct point in $E$.
Perfect: If $E$ is closed and if every point of $E$ is a limit point of $E$.
I'm not quite sure I understand the difference between a point and a limit point…
This reminds me, since the complement of an open set is closed, does that mean that the complement of the complex plane, the empty set, is neither open nor closed ?
Best Answer
The empty set and the whole space are open and closed at the same time. A limit point is a point that does not belong to the given set, but belongs to some super set (otherwise it makes no sense). The whole plane is open because every point is interior (it has no frontier). It is closed, because it contains all the points, in particular, the limit points. Finally, it is perfect, because any point is the limit point: Take any point, in any neighborhood of it, there are infinitely many points of the plane.