@Charles
Yup. A topologist would say that a there is a homomorphism between (T^2 = (S1)^2) \ {0, 0} and R^2, and that there was a homomorphism between any sufficiently small neighborhood in T^2 and the open disk in R^2.
A differential geometer (or a physicist) would counter that there was no diffeomorphism between T^2 and any subset of R^2, so sufficiently accurate measurements of curvature in even a fairly large T^2 would reveal the non-Euclidean nature of the space.