I am trying to do the following exercise :
A $C^1$ immersion $f:M\rightarrow N$ which is injective on a closed subset $F\subset M$ is injective on a neighborhood of $F$. In fact $f$ has a neighborhood $ V\subset C_S^1(M,N)$ and $F$ has a neighborhood $U\subset M$ such that every $g\in V$ is injective on $U$. If $K$ is compact $V$ can be taken in $C_W^1(M,N)$.
Focusing only on the first part of the problem that is the part where can extend injectivity of closed sets ininjective immersions. I have been thinking about this for a while and there are two ideas, either I construct the neighborhood , I have tried this using the fact that $f$ is locally injective , or I go by contradiction but I can't get anytihing. Either way I got nowhere useful and was wondering if there is anything else that I can try or if there is something I am missing ?
Any advice on this is aprecciated. Thanks !
Hint: Proper maps are closed maps; immersions are locally injective. Then try to construct the neighborhood on a countable series of ordered compact subsets , with induction.