Local degree of local homeomorphism is $\pm 1$

Question

Let $f:X\to Y$ be a local homeomorphism. I claim that local degree of $f$ is $\pm 1$. I was wondering if my proof is correct: Let $x\in f^{-1}(\{y\})$ , $U$ be a neighbourhood of $x$ and $V$ be a neighbourhood of $y$ so that $f\vert_{U}:U\to V$ is homeomorphism. Then $f_{*}:H_{n}(U,U-\{x\})\to H_{n}(V,V-\{y\})$ is an isomorphism. Hence local degree at $x$ is $\pm1$. Is the proof OK?

Answer 1

The proof is correct; however, remember to add the assumption that both $X$ and $Y$ are manifolds. As Idrissi says in the comments, this proof also applies to homology manifolds.