Topology and mapping of boundary points Suppose $\Omega$ is a set of points in the xy plane, and f is a continuous transformation from points in the xy plane to points in the xy plane.
What are the necessary and sufficient conditions for this:
$f(\partial\Omega)\subset \partial(f (\Omega))$
ie: boundary points map to boundary points.
 A: I am assuming that by "$\subset$", you mean "$\subseteq$".
We have
$$
f(\partial \Omega) \subseteq \partial f(\Omega)
\iff
f(\partial \Omega) \subseteq \text{cl } f(\Omega)
\land
f(\partial \Omega) \cap \text{int } f(\Omega) = \emptyset
$$
Because $f$ is continuous, we have $f(\text{cl } \Omega) \subseteq \text{cl } f(\Omega)$, so
$$
f(\partial \Omega) \subseteq \partial f(\Omega)
\iff
f(\partial \Omega) \cap \text{int } f(\Omega) = \emptyset
\\ \iff
(\forall x \in \text{cl } \Omega : f(x) \in \text{int } f(\Omega) \implies x \in \text{int } \Omega)
\\ \iff
\text{cl } \Omega \cap f^{-1}(\text{int } f(\Omega)) \subseteq \text{int } \Omega
$$
(If any of the steps are unclear, please ask me!)
Now I can't simplify this any further (does anybody else have an idea?).
Also, I could only think of one sufficient condition:


*

*If $f$ is injective, your condition holds, since we have $f^{-1}(f(\Omega)) = \Omega$, but since $f$ is continuous, the preimage is also open, so $f^{-1}(\text{int } f(\Omega)) \subseteq \text{int } \Omega$.

