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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 :).

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  • $\begingroup$ How are you defining the boundary here? $\endgroup$ – copper.hat Sep 2 '14 at 21:18
  • $\begingroup$ When you say "$X$ is a bounded open set", do you mean that $X\subset \mathbb R^n$? In other words, in what context do you mean "bounded"? $\endgroup$ – MPW Sep 2 '14 at 21:22
  • $\begingroup$ I guess the OP means something like this: Let $X$ and $Y$ be topological spaces and $f:X\to Y$ a continuous open map. Is it then true that $\partial f(A)\subset f(\partial A)$ for every set $A\subset X$? Or for every open set $A\subset X$? $\endgroup$ – Joonas Ilmavirta Sep 2 '14 at 21:23
  • $\begingroup$ if $W$ is open bounded set, we say that $x\in \partial W $ if for every neighborhood $U$ of $x$ we have that $U\cap W \neq \oslash$ and $U\cap int(W^c) \neq \oslash$ $\endgroup$ – Porufes Sep 2 '14 at 21:26
  • $\begingroup$ You need more info. The boundary of $X$ is empty in this case... $\endgroup$ – copper.hat Sep 2 '14 at 21:33
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No. Take $f: (0,1) \to (0,2)$ defined by $f(x) = x$. This is a continuous open function, but $\partial X = \emptyset$ while $\partial f(X) = \{1\}$.

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