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^{-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?