Topological Invariance of Surfaces of Constant Curvature in Euclidean 3-space

We explore the topic of topological invariance by investigating aspects of Gauss-Bonnet Theorem related to surfaces of constant curvature embedded in Euclidean 3-space. We informally show that closed, compact, simply connected 2-manifolds without boundary in Euclidean 3-space are all homeomorphic to the sphere.