I want to know the mathematical term for connected region the boundary of which is connected, if exists.
Intuitively, I want to define a connected region which does not have any hollow inside. For example, in the figure below, the first one is OK as it is connected and also does not have any hollows inside. The second one is not OK as it is not connected. The third one is not OK as it has a hollow so the boundary of the region is not connected.
A region without "holes" is usually called SIMPLY CONNECTED, but this is NOT the same as you ask fo: For exmaple, think on the exterior of a circle: it is a connected region and its boundary is connected, BUT, this is not a simply connected region.