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Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their one-point compactifications and let $\mathrm{map}(\bar{V},\bar{U})$ denote the space of maps between them again with the compact-open topology. There is a (set-theoretic) map $$\mathrm{OEmb}(U,V)\longrightarrow \mathrm{map}(\bar{V},\bar{U})$$ that sends an embedding $e$ to the map $\phi(e)$ given by $\phi(e)(y) = e^{-1}(y)$ if this value exists and $\phi(e)(y)=\infty$ otherwise.

Fact: This map is continuous. More generally, it is also continuous if $U$ and $V$ are Hausdorff locally compact locally connected topological spaces.

What reference should I cite for these facts?

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  • $\begingroup$ I suppose it depends on your purpose. Are you posting the references for a MO question? Perhaps for a paper? A blog? Some context would be helpful. $\endgroup$ – Ryan Budney Dec 9 '14 at 19:22
  • $\begingroup$ @Ryan The purpose is to use the result in a paper. $\endgroup$ – Federico Cantero Dec 9 '14 at 19:47
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    $\begingroup$ It looks quite similar to Segal's scanning map from his 1973 paper on configuration spaces and iterated loop spaces. $\endgroup$ – Ryan Budney Dec 10 '14 at 17:22
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I haven't found an explicit reference in the literature, but as Johannes Ebert pointed to me, in the second part of Theorem 4 of

R. Arens, Topologies for Homeomorphism Groups, American Journal of Mathematics, Vol. 68, No. 4 (Oct., 1946), pp. 593-610

it is proven that the inverse mapping $\mathrm{Homeo}(X)\to \mathrm{Homeo}(X)$ that sends a homeomorphism to its inverse is continuous with respect to the compact-open topology, provided that $X$ is Hausdorff, locally compact and locally connected. The same proof works as well to prove my question (under the assumptions that $X$ and $Y$ are Hausdorff, locally compact, and $Y$ is locally connected). Therefore I think that Arens article is a good reference for this fact (together with an indication saying that the proof has to be mildly adpated). This is how the adaptation would go:

Theorem: If $X$ and $Y$ are Hausdorff locally compact spaces and $Y$ is locally connected, then the collapsing map $$\phi\colon \mathrm{OEmb}(X,Y)\longrightarrow \mathrm{map}(\overline{Y},\overline{X})$$ is continuous with the compact-open topologies.

Proof: Take an embedding $e$ and consider a subbasic neighbourhood $(K,U)$ of $\phi(e)$, where $K\subset \overline{Y}$ is compact and $U\subset \overline{X}$ is open. Let us denote by $U^c$ the complement of $U$ in $X$ (not in $\overline{X}$). An embedding $f$ belongs to this neighbourhood if $\phi(e)(K)\subset U$, and this holds if and only if $e(U^c)\subset K^c$. Therefore $e(U^c)\subset K^c$ and if any other embedding $f$ satisfies that $f(U^c)\subset K^c$, then it holds that $\phi(f)\in (K,U)$.

We distinguish two cases:

\underline{If $\infty\in U$}, then $U^c$ is compact and we are done ($(U^c,K^c)$ is an open neighbourhood of $e$ that gets mapped into the neighbourhood $(K,U)$ of $\phi(e)$.

\underline{If $\infty\notin U$}, then $K\subset e(X)$, and therefore $e^{-1}(K)$ is compact. This is the situation that Arens handles. We use that $X$ is Hausdorff locally compact to construct a pair of open neighbourhoods $V,W$ of $K$ with compact closure of $e$ such that $$e^{-1}(K)\in V\subset \overline{V}\subset W\subset \overline{W}\subset U.$$ Then $N:=\overline{W}\setminus V$ is compact, and if $f\in (N,K^c)$, then $\phi(f)(K)\subset N^c$. Observe that $N^c$ is the disjoint union of $V$ and $\overline{W}^c$.

Now we use that $X$ is locally connected to assume, without loss of generality that $K$ is connected (this is detailed in Arens' paper). Therefore, if $f\in (N,K^c)$, then either $\phi(f)(K)\subset V\subset U$ (this is what we want) or $\phi(f)(K)\subset \overline{W}^c$. In order to rule out the second case, we make the additional assumption (again without loss of generality) that $K$ has non-void interior, then we take a point $p$ in the interior. If $f\in (N,K^c\cap e(W))\cap (\{e^{-1}(p)\},\mathring{K})$, then $f(e^{-1}(p))\subset K$, but then $e^{-1}(p)\in \phi(f)(K)\cap V$, which is non-empty, and therefore we have ruled out the second case and $f\in (K,V)\subset (K,U)$.

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