  We derive the most general expression for the Skyrmion topological charge for
a two-dimensional spin texture, valid for any type of boundary conditions or
for any arbitrary spatial region within the texture. It reduces to the usual
one $Q = 1/4\pi \iint \vec f \cdot \left(\partial_x \vec f \times \partial_y
\vec f\right)$ for the appropriate boundary conditions, with $\vec f$ the spin
texture. The general expression resembles the Gauss-Bonet theorem for the
Euler-Poincar\'e characteristic of a 2D surface, but it has definite
differences, responsible for the assignment of the proper signs of the Skyrmion
charges. Additionally, we show that the charge of a single Skyrmion is the
product of the value of the normal component of the spin texture at the
singularity times the Index or winding number of the transverse texture, a
generalization of a Poincar\'e theorem. We illustrate our general results
analyzing in detail a two-component spinor Bose-Einstein condensate in 2D in
the presence of an external magnetic field, via the Gross-Pitaevskii equation.
The condensate spin textures present Skyrmions singularities at the spatial
locations where the transverse magnetic field vanishes. We show that the
ensuing superfluid vortices and Skyrmions have the same value for their
corresponding topological charges, in turn due to the structure of the magnetic
field.
