Answer
a) Yes
b) Yes
c) No
Work Step by Step
a) For the open set, the region should not contain any boundary poins. The set contains all points in the xy plane except the point $(2,3)$. There is no point on the boundary which belongs to the set. When we draw a disk for the given set, we find that it entirely lies inside the region $D$. Thus, the set is open.
b) For the connected set, any two points in the region $D$ can be connected by a path that lies entirely in $D$. The set contains all points in the xy plane except the point $(2,3)$. From the given points, we can draw a path connecting the two points in $D$ without passing through $(2,3)$. Thus, the set is connected.
c) For the simply connected set, the region must not have any holes or be divided into two parts.The set contains all points in the xy plane except the point $(2,3)$. From the given points, it has been seen that the path connecting the two points does not completely lie inside the given set because it passes through the point $(2,3)$, which is not in the set. Thus, the set is not simply connected.
