# Metric spaces - sets proof

Let $(X,\rho)$ be a metric space and $A\subset X$. Prove that $\partial A = \overline A \cap \overline{X \setminus A}$. I have no idea how to prove that. Please help.

• How is teh boundary defined? As the closure less the interior? – ncmathsadist May 28 '13 at 23:58
• Not every question including the word "set" is a question in set theory. – Asaf Karagila May 28 '13 at 23:58
• What definition of $\partial A$ are you using? Have you solved other problems where you have to show that two sets are equal? – Jonas Meyer May 28 '13 at 23:59
• C'est vrai, Asaf. – ncmathsadist May 28 '13 at 23:59
• My definition is that $\partial A$ is the set of all points all of whose neighborhoods meet both $A$ and $A^c$. – ncmathsadist May 29 '13 at 0:00

1. Let $x \in \partial A$ arbitrary. Show that $x \in \overline{A}$ (this should be easy from your definition). Then, show that $x \in \overline{X \setminus A}$ using the fact that $\partial A = \partial (X \setminus A)$.
2. Let $y \in \overline{A} \cap \overline{X \setminus A}$ arbitrary. Show that $y \in \partial A$ by showing that $y \in \overline{A}$ but $y \not \in \text{Int } A$.
• "using the fact that $\partial A=\partial(X\setminus A)$": Also proving that fact, of course:) – Jonas Meyer May 29 '13 at 0:40