This question regards Guibas and Stolfi's Primitives for the manipulation of general subdivisions and the computation of Voronoi, specifically their statements on page 77. In this paper, the authors define a 'face' of some partition, S, of a manifold, M, as an element of S that is homeomorphic to a 2D open disk. They later state:
... Therefore the closure of a face may not be homeomorphic to a disk, as Figure 1 shows.
I think I understand this latter statement, but I am unsure, so I would like to check my understanding.
Figure 1 displays three subdivisions. I believe the top two are of a sphere, while the bottom subdivision is of a torus (although I am uncertain about this, given that the paper heavily focuses on M being a sphere...). I think the referred example is for the torus, for which I've copied their figure (left) and colored it (right) here: Annotated 'bottom' subdivision from Figure 1 of Guibas and Stolfi's Primitives for the manipulation of general subdivisions and the computation of Voronoi.
Here, I think the failure is that the interior of the 'purple' region is homeomorphic to a disk, but its closure is not since two separated 'purple' regions both 'touch' the brown vertex. Is that correct? If not, could I please get some clarification?