Let $U$ be a connected open subset in $\mathbb{R^n}$. Let $f: U \rightarrow U$ be a differentiable projection, i.e. $f\circ f = f$. It's well-known that $f(U)$ is a submanifold of $U$ (Henri Cartan, *Sur les rétractions d’une variété.(1986)*. My question is

*"Under which condition on $f$ (and possibly on $U$) we have $Cl_{\mathbb{R}^n} f(U) \cap \partial_{\mathbb{R}^n} U \neq \emptyset$?"*

Here the notation $Cl_{\mathbb{R}^n}$ stands for the closure when considering as a subset of $\mathbb{R}^n$ and the notation $\partial_{\mathbb{R}^n}$ stands for the topological boundary as a subset of $\mathbb{R}^n$

For example, if we take $U= \{ x \in \mathbb{R}^n | \|x \| <1\}$ the open solid sphere and the map $f(x_1,\ldots,x_n)=(0, x_2,\ldots,x_n)$ the projection onto the plane $\{x_1=0\}$. Then $f(U)$ is the unit disk in the plane $\{x_1=0\}$ and its closure as a subset of $\mathbb{R}^n$ is $$\{x=(x_1,\ldots,x_n) \in \mathbb{R}^n |x_1=0, \sum\limits_{i=2}^n x_i^2 \le1\}$$ indeed has non empty intersection with the boundary of $U$.

Further more, I am wondering if we could replace the hypothesis by $U$ is in $\mathbb{C}^n$ and the map $f$ is holomorphic retraction. Is that more helpful if we add condition on the boundary of $U$ (a submanifold, a algebraic subset)?

6more comments