Answer
a) Yes
b) Not
c) Not
Work Step by Step
a) The region should not have any boundary point in order to get the open set . There is no boundary point in the given set $1\lt |x| \lt 2$ on the set $y$ and also, the given set has only value in $R$.So, the set is open.
b) Any two points in the region domain $D$ should be connected to a straight segment path In order to get the connected set that lies entirely inside the domain $D$. So, the set is connected. From the given points we cannot draw a path without crossing the boundary, so, will have to move outside the set. However, the given is NOT connected.
c) The region should not contain any holes or does not divided into two parts In order to get the simply connected set But the given points that the path has been joined the two points could not completely lie inside the given set. So, the given set is not simply connected.
