Answer
a) Yes
b) Yes
c) Yes
Work Step by Step
a) In order to get the open set, the region should not have any boundary points. As we can see from our set $0\lt y\lt 3$, there is no boundary point.
Hence, the set is open.
b) In order to get the connected set, any two points in the region $D$ must be joined by a straight segment path that lies entirely inside the domain $D$. This is the case for our set.
Hence, the set is connected.
c) In order to get the simply connected set, the region should not contain any holes or get divided into two parts. Thus, the given set $0\lt y\lt 3$ satisfies.
Hence, the set is simply connected.
