Image of Boundary of Half Closed plane under Diffeomorphism Let $S: A\to B$ be diffeomorphism, where A and B are an open subset of closed half-space in $R^n= \{x\in \mathbb R^n|x_1\geq 0\}$.$\partial H=\{x\in R^n|x_1=0\}$. Then image of $\partial H \cap A $ is $\partial H \cap B $ under S.


It is clear that $\partial H \cap A $ does not contain any interior point. I thought to use the inverse function theorem.

Suppose on contrary it maps to some interior point. Then there exist nhbd of that point which is homeomorphic to the inverse image of the map under restriction. which is a contradiction to the fact that domain does not contain any open nbhd.

Is this argument is enough? 

Any help will be appreciated.
which is a contradiction to the fact that domain does not contain any open nbhd.


What do you mean by that? Clearly the domain, being $\partial H$ contains lots of open neighbourhoods. No, this is not enough.

One of the possible solutions follows from the homotopy theory. So assume that there is a homeomorphism $S:A\to B$ that maps an interior point to a boundary point. In other words we have a homeomorphism $f:D\to C$ where $D$ is an open Euclidean ball in $\mathbb{R}^n$ and $C$ is some open subset of $\mathbb{R}^n_{\geq 0}$ that has a nonempty intersection with boundary $\partial H$. Note that $D$ is contractible and therefore so is $C$. But $D\backslash\{p\}$ is not contractible for any point $p\in D$ (it is a homotopy sphere). However $C\backslash\{q\}$ remains contractible when $q$ belongs to the boundary. This can be derived from the fact that $C'\backslash\{q\}$ is a star domain where $q$ is boundary point and $C'$ and open ball around $q$ in $\mathbb{R}^n_{\geq 0}$.
which is a contradiction to the fact that domain does not contain any open nbhd.


What do you mean by that? Clearly the domain, being $\partial H$ contains lots of open neighbourhoods. No, this is not enough.

One of the possible solutions follows from the homotopy theory. So assume that there is a homeomorphism $S:A\to B$ that maps an interior point to a boundary point. In other words we have a homeomorphism $f:D\to C$ where $D$ is an open Euclidean ball in $\mathbb{R}^n$ and $C$ is some open subset of $\mathbb{R}^n_{\geq 0}$ that has a nonempty intersection with boundary $\partial H$. Note that $D$ is contractible and therefore so is $C$. But $D\backslash\{p\}$ is not contractible for any point $p\in D$ (it is a homotopy sphere). However $C\backslash\{q\}$ remains contractible when $q$ belongs to the boundary. This can be derived from the fact that $C'\backslash\{q\}$ is a star domain where $q$ is boundary point and $C'$ and open ball around $q$ in $\mathbb{R}^n_{\geq 0}$.