Let us say a given topological (commutative) ring $K$ has property $P$ if for each $f \in x + (x^2)(K[x])$, such $f$ is injective on some neighborhood of $0 \in K$ when viewed as a function $f : K \rightarrow K$.
From what I understand:
- Any topological subring of a topological commutative ring with property $P$ also has property $P$.
- The field of complex numbers has property $P$.
- According to this answer to my earlier question, any topological field which is complete with respect to a non-archimedean absolute value has property $P$. (E.g. the $p$-adic number fields.)
I was wondering, what would be an example of a Hausdorff topological field which does not have property $P$?