[Math] Image of boundary of continuous open function

Question

Edit to be more clear:

Let X and Y be topological spaces and $f:X→Y$ a continuous open map. Is it then true that $∂f(A)⊂f(∂A)$ for every open $A⊂X$ such that $∂A \neq \oslash$.

thanks for your help :).

Answer 1

No. Take $f: (0,1) \to (0,2)$ defined by $f(x) = x$. This is a continuous open function, but $\partial X = \emptyset$ while $\partial f(X) = \{1\}$.