So I'm having trouble understanding how these two are related, i.e., how one proves the other. I understand the ideas behind both of them: For J-B, you're basically taking R^n and throwing in a sphere, so the inside of the sphere is bounded and everything outside the sphere is unbounded. For Invariance of Domain, it's pretty obvious just by the definition (the image of an open subset of R^n is open). However, I don't really see a relationship between the two. Can anyone give some insight? Also, I was curious why we only need injectivity for both of them, where surjectivity is basically unnecessary in any possible proof. What makes 1-1 important?