How to determine whether those sets are open or closed? Given those three sets below, A (left), B (center) and C (right), with A, B, C $\subseteq \mathbb{R^2}$, how can I determine, whether they are open or closed in metric space terminology via simplest way?
Set A is a closed set, set B is open set and set C is a closed set with a hole with boundary in center of it.

 A: A set, in a metric space, is "open" if and only if it contains none of its boundary points.  If I am correct in interpreting the "dashed" boundary on the second square as meaning that those points are not in the set, then that set is open.
A set, in a metric space, is "closed" if and only if it contains all of its boundary points.  If I am correct in interpreting the "solid" boundary on the first and third sets as meaning that those points are in the set, then that set is closed.
Of course, if a set contains some but not all of its boundary points then it is neither open nor closed.
It is even possible for a set to be both open and closed if it has no boundary points.  In $\mathbb{R}^2$, the only such sets are the empty set and $\mathbb{R}^2$ itself.
A: A set is called open if every point of s has a neighborhood that consists of entirely from points that belong set S. Now think about it, if there is no boundary, we can find new points by just magnifying it, there are infinitely many of them. But if there is boundary, then choose a point on boundary. The neighborhood of this points has some points that does not belong to original set.
