I read a little introductory book to topology. It basically said that for any two-dimensional manifold (well maybe just the closed ones, as I think about it) its topology can be unambiguously defined by the Euler's characteristic and orientability.
Consequently it is very nice that any convex polyhedron has its Euler characteristic $\chi=2$ which means that the global topology is the same as for the sphere.
What about the examples below? In fact $\chi>2$ which occurs in these cases do not correspond to any global topologies I know.
If you have a space made up of triangles (a 2 dimensional simplicial complex, if you will, although this generalizes to higher dimensions as well) the Euler characteristic is just #vertices - #edges + #triangles. The main theorem is that this number is only dependent on the underlying topological space (up to homotopy) and not on how you go about triangulating it, this is a corollary of the fact that simplicial and singular homology theories agree. In both of these examples the underlying space is homotopy equivalent to two spheres connected at a point, so as expected they have the same Euler characteristic (3).