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I am reading T. Szamuely's book on Galois groups and fundamental groups. As preparation to the algebraic case, he recalls the topological case. So I am wondering if a surjective local homeomorphism $f$ from some connected space $X$ to $\mathbb{R}^n$ is necessarily a covering, in which case it would be bijective since $\mathbb{R}^n$ is simply connected.

What about the differentiable case?

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  • $\begingroup$ Local on the source or on the target? $\endgroup$
    – Emerton
    Commented Jan 30, 2010 at 16:46
  • $\begingroup$ "numerical space" -> "Euclidean space" $\endgroup$
    – S. Carnahan ♦
    Commented Jan 30, 2010 at 17:50
  • $\begingroup$ Strictly speaking, I think Euclidean space has less structure than R^n, so I'd be more inclined to call the latter "Cartesian space"... $\endgroup$
    – Darsh Ranjan
    Commented Jan 30, 2010 at 22:03

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