Answer
a) Yes
b) Not
c) Not
Work Step by Step
a) For the open set, the region should not contain any boundary points. In the given set, we have $1\lt |x| \lt 2$, in which we have no condition on the y set. This means that y can have only value in $R$ and the set does not contain any boundary points. 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 the region $D$. From the given points we cannot draw a path without crossing the boundary, so we will have to move outside the set. Thus, the set is not connected.
c) For the simply connected set, the region must not have any holes or be divided into two parts. From the given points, it has been seen that the path connecting the two points does not completely lie inside the given set. Thus, the set is not simply connected.
