  In this letter, we use a recent wormhole solution known as a ringhole
[Gonzalez-Diaz, Phys.\ Rev.\ D {\bf 54}, 6122, 1996] to determine the surface
topology and the deflection angle of light in the weak limit approximation
using the Gauss-Bonnet theorem (GBT). We apply the GBT and show that the
surface topology at the wormhole throat is indeed a torus by computing the
Euler characteristic number. As a special case of the ringhole solution, one
can find the Ellis wormhole which has the surface topology of a 2-sphere at the
wormhole throat. The most interesting results of this paper concerns the
problem of gravitational deflection of light in the spacetime of a ringhole
geometry by applying the GBT to the optical ringhole geometry. It is shown
that, the deflection angle of light depends entirely on the geometric structure
of the ringhole geometry encoded by the parameters $b_0$ and $a$, being the
ringhole throat radius and the radius of the circumference generated by the
circular axis of the torus, respectively. As special cases of our general
result, the deflection angle by Ellis wormhole is obtained. Finally, we work
out the problem of deflection of relativistic massive particles and show that
the deflection angle remains unaltered by the speed of the particles.
