Is a $C^1$ immersion that is injective on a closed set $K$ injective on a neighborhood of $K$?
I'm doing problem 7 of section 2.1 on "Differential topology" written by Hirsch which goes as $$\begin{array}{l}{\text { 7. A } C^{1} \text { immersion } f : M \rightarrow N \text { which is injective on a closed subset } K \subset M \text { is injective }} \\ {\text { on a neighborhood of } K . \text { In fact } f \text { has a neighborhood } \mathscr{N} \subset C_{S}^{1}(M, N) \text { and } K \text { has a }} \\ {\text { neighborhood } U \subset M \text { such that every } g \in \mathscr{N} \text { is injective on } U . \text { If } K \text { is compact } \mathcal{N}} \\ {\text { can be taken in } C_{W}^{1}(M, N) .}\end{array}$$ I have no problem about the compact case, but I think the conclusion does not hold when $$K$$ is not compact for the following counterexample that I cook up.
Consider the figure "8" $$\beta :(-\pi, \pi) \rightarrow \mathbb{R}^{2}, \text { with } \beta(t)=(\sin t, \sin 2t)$$ and $$n(t)$$ be its normal $$n(t)=(-2\cos2t,\cos t)$$. We define $$f:(-\pi, \pi)\times(-1,1)\to\mathbb{R}^{2}$$ $$f(t,s)=\beta(t)+sn(t)$$. Then $$f$$ is $$C^1$$ immersion which is injective on a closed subset $$K=(-\pi,\pi)\times\{0\}$$ but $$f$$ cannot be injective on a neighborhood of $$K$$ since $$f$$ is not injective on $$K\cup B_\delta(0)$$ for any $$\delta>0$$.