The definition of a homeomorphism between topological spaces X, Y, is that there exists a function Y=f(X) that is continuous and whose inverse X=f-1(Y) is also continuous. Can I assume that the function f is a bijection, since inverses only exist for bijections? Also, I thought that if a function f is continuous, then its inverse is automatically continuous? So why is there a need to mention that the inverse is also continuous in the definition I have above? Also, a manifold is defined to be a topological space that is locally homeomorphic to R^n. So take the interval [a,b] on R. Would this be a manifold? [a,b] is compact so is not homeomorphic to R, but locally it can be done with an open covering of [a,b]. If there exists just one open covering where each of the open sets is homeomorphic to R^n, then is that sufficient to say the space is a manifold?