Relative Topology in $\mathbb{R}^n$ I'm studying relative topology for the first time and I've come across a proposition which states if $V \subset \mathbb{R}^n$ is non empty and $A \subseteq V$ $$ \partial_v A = \partial A \cap V.$$  The proposition is clearly false if you take $A=V$ and $V$ closed then, $\partial_v V \subseteq \text{cl}_V(V - V)=\emptyset.$ Whereas, $\partial V$ is certainly not empty. 
The following is a proof on someone elses question Boundary of set on relative topology in $R^n$. This doesn't work either, I've proven it for the case that $V$ is open, is that the only case that it holds for all $A \subset V$ (I know it is for $A \subseteq V$)? 
 A: Denote closure in $V$ by $Cl_V$ and closure in the ambient space $X$ (here $X = \mathbb{R}^n$) by $Cl_X$. Similarly for interior by $Int_V, Int_X$. Then $\partial_V A = Cl_V(A) - Int_V(A)$ and $\partial_X A = Cl_X(A) - Int_X(A)$. It is then sufficient to prove your proposition that $Cl_V(A) = Cl_X(A) \cap V$ and $Int_V(A) = Int_X(A) \cap V = Int_X(A)$.
$Cl_V(A)$ is precisely $A$ union its limit points in $V$. Thus it is sufficient to show that the set of limit points of $A$ in $V$ is equal to the set of limit points of $A$ in $X$ intersect $V$. The forward inclusion $\subseteq$ is clear. Suppose $x \in V$ is a limit point of $V$ in $X$. Let $U$ an open set in $V$ containing $x$. Then $U = B \cap V$ where $B$ is an open set in $X$ containing $x$. Then $B \cap A \backslash \{x\} \neq \emptyset$ as $x$ is a limit point of $A$ in $X$. Then $A \subseteq V$ so $U \cap A \backslash\{x\} = V \cap B \cap A \backslash\{x\} = B \cap A \backslash \{x\} \neq \emptyset$. So $x$ is a limit point of $A$ in $V$, so we get the backwards inclusions $\supseteq$. Thus $Cl_V(A) = Cl_X(A) \cap V$.
Edit: I realize I've made a mistake. The proposition certainly doesn't hold in general, since in general $Int_V(A) \supseteq Int_X(A)$ (as can be easily checked), but the reverse containment doesn't hold. So the best you can hope for in general is $\partial_V A \subseteq \partial_X A$
Edit 2: I just looked at the proof in the book OP mentioned. The authors make the argument:
$$\partial_V A = (V \cap Cl_{\mathbb{R}^n}(A)) \cap (V \cap Cl_{\mathbb{R}^n}(V \backslash A)) = (V \cap Cl_{\mathbb{R}^n}(A)) \cap (V \cap Cl_{\mathbb{R}^n} ({\mathbb{R}^n} \backslash A)) = \cdots$$
But this is wrong: $V \cap Cl_{\mathbb{R}^n}(V \backslash A)) \neq V \cap Cl_{\mathbb{R}^n} ({\mathbb{R}^n} \backslash A)$ in general. E.g. if $V$ is the closed unit ball.
