Topology question: are surfaces, and surfaces with boundary, mutually exclusive? The textbook wording was a bit ambiguous. Consider for example a sphere. This is Hausdorff, and all points can be contained within an open set which is locally homeomorphic to an open disk in the plane. But the book defines a surface with boundary as having open sets homeomorphic to an open disk OR a half disk. For the sphere, every point that can have an open disk surrounding it could also have a half disk (although it doesn't need to). By this definition, is a sphere both a surface AND a surface with boundary? Are all surfaces also surfaces with boundary? I know not all surfaces with boundary are surfaces (see: semisphere) but not sure about the other way around.