Differential Geometry: Lecture 14 part 1: topological trivia for surfaces

here we take a quick tour of the major toplogical terms for surfaces and we find some nice improvements to previoua hand-waving in multivariate calculus. In particular, the concept of homotopy and integration of forms over two-segments gives some quantitative measure of simple connectivity (in previous courses you probably just said something about loops shrinking to points, this makes that a bit more precise). I don't prove much here as I leave that to the topology course :) (or Oneill as his sectino 4.7 is quite complete for our purposes) In the next part I give the proof I gave up on for the sake of dinner and I say a word or two about manifolds (we say more much later in the course)