I recently tried to wrap my head around the following problem: Let $f\colon \mathbb{R} \times K \rightarrow \mathbb{R}$ be a smooth map, where $K$ is a compact manifold. Assume that for each $k\in K$, the partial map $f(\cdot,k) \in \mathrm{Diff}(\mathbb{R})$ (the group of smooth diffeomorphisms), so in particular for every fixed $k$ the partial map is proper (i.e. Preimages of compact sets are compact).


**Question**: Is it then true that $f$ is a proper map?


Note that the smoothness of $f$ is probably inessential and one could ask the same question for continuous maps taking their image in homeomorphisms of the real line.


The reason I am asking is that these mappings appear in the context of the Newman-Unti group from general relativity. There they arise as one component of certain diffeomorphisms of $\mathbb{R} \times K$, so in particular the $f$ generated that way will be proper. However, physics literature only states the above condition on the partial mappings $f(\cdot, k)$. So i wondered if properness is a consequence of the above assumptions or must be added as a requirement



Parametrised proper map
Hei Ryan, thanks for the comment, now I feel really stupid (if you post it as an answer I will accept it). Here is now the full argument (which is actually quite straight forward):


Let $L$ be a compact subset of the reals and $F \colon \mathbb{R} \times K \rightarrow \mathbb{R}$ be a continuous mappping such that the partial maps $f(\cdot,k)$ are homeomorphisms of the reals for all $k \in K$.


To see that also $F^{-1}(L)$ is compact, we pick a sequence $(r\_\ell , k\_\ell)$ in the preimage. Then by compactness of $L$, we can pass to a subsequence such that $F(r\_\ell, k\_\ell)$ converges in $L$ towards a limit, say $z$. By compactness of $K$ we pass again to a subsequence such that $k\_\ell$ converges to a limit $k\_\ast \in K$. Note that since $F(\cdot, k\_\ast)$ is a homeomorphism we find $r\_\ast = F(\cdot , k\_\ast)^{-1} (z)$ and claim now that the sequence $r\_\ell$ converges to $r\_\ast$.


To see this note that we have a sequence of homeomorphisms
$$G\_\ell := F(\cdot, k\_\ast)^{-1} \circ F(\cdot, k\_\ell) , \quad \ell \in \mathbb{N} $$
which converges (pointwise) towards the identity and by construction $G\_\ell (r\_\ell) \rightarrow r\_\ast$. Or in other words $r\_\ell - G\_\ell^{-1}(r\_\ast)$ converges to $0$, since also $G\_\ell^{-1}(r\_\ast) \rightarrow r\_\ast$ we see that $r\_\ell \rightarrow r\_\ast$. This proves compactness.


Again thanks Ryan for the nod in the right direction. Now its an undergraduate exercise.


