Local homeomorphism which is one-to-one in a closed is a homeomorphism in a neighborhood of this set

Question

Let $M, N$ be Hausdorff spaces, $M$ locally compact with a countable basis, $\phi\colon M \to N$ a local homeomorphism, $A \subset M$ closed in $M$ and $\phi|_A$ one-to-one. Then there exists a neighborhood $V$ of $A$ in $M$ such that $\phi|_V$ is a homeomorphism.

The above is an unproven claim in I. Chavel, Riemannian Geometry: A Modern Introduction [doi:10.1017/CBO9780511616822] which appears at the very end of the proof of Theorem I.3.2.

I was wondering if someone could lend me a hand in showing this result.

Answer 1

Chavel's claim is false as stated.

Consider $N=R^2$, $A\subset R^2$ a 1-dimensional linear subspace, $M$ a suitable open neighborhood of $A$ in $R^2$ (hence, $M$ is also homeomorphic to $N$). There exists an injective continuous map $f: A\to N$ which is the identity near $0$ and which maps $A$ to the "figure 8" curve. (A picture would be good here, I will add it when I have time.) Then map $f$ admits a locally homeomorphic extension to a map $F: M\to N$. However, there is no open neighborhood $V$ of $A$ in $M$ such that $F|V$ is 1-1.

See also this question (I am sure, it was asked even earlier).