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Let $f:M\to N$ be continuous and locally injective. If M is connected and exist a continuous function $g:N\to M$ such that $ fg= id_N$ then f is a homeomorphism from M to N.

First clearly f is surjective, since exist a right inverse. But I don't know how to proceed. I don't know how to prove the injectivity.

This problem it's from a book of metric spaces, maybe the hyphotesis also holds for general topological spaces, I'm not sure.

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    Dear Arkj,$I$would start with fixing a point $n\in N$ and consider an open neighbourhood $I$ of $g(n)$ on which $f$ is injective. By suitably shrinking $I$ to $I'$ I would try to show that $g$ restricts to a homeomorphism from $g^{-1}(I')$ to $I'$ (with inverse $f$ of course). The difficulty is to keep a precise book-keeping of the shrinkings...2012-05-02

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