Chapter 16 - Vector Calculus - 16.3 The Fundamental Theorem for Line Integrals - 16.3 Exercises - Page 1135: 34

a) Yes b) Yes c) No

a) The region should not have any boundary point in order to get the open set . The given set consists of all points in the xy plane except the points lie at $(2,3)$, this means that there is no boundary points on the set. Also, when we draw a disk for the given set , it has been noticed that the set lie inside the region $D$. So, 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 the $D$.he given set consists of all points in the xy plane except the points lie at $(2,3)$, this means that there is no boundary points on the set. With these points when we can draw a path we see that it connects the two points in the $D$ passes through the point $(2,3)$. So, the set is connected. c) For the simply connected set, the region must not have any holes or divided into two parts.he given set consists of all points in the xy plane except the points lie at $(2,3)$, this means that there is no boundary points on the set and the path does not join the two points completely and lie inside the set .Because it passes through the points $(2,3)$ which is not in the set. So, the set is Not simply connected.