Let $(X, mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) cap Cl(X-A)$.
I know this is a true statement. I am trying to prove if because I would also like to use this definition to show that the boundary of a set is a closed set.
Here are my definitions, knowledge and attempt at the proof. I am using the definitions and set theory to complete my proof by letting an element be in each side and then show it is in the other side.
Boundary of $A$ is the set of all points $x in X$ for which every open set containing $x$ intersects both $X$ and $X-A$
Closure of $A$ is $Cl(A) = bigcap {U subseteq X: U$ is a closed set and $A subseteq U}$. I know from my definition that $A subset Cl(A)$
My attempt at the proof:
Let $x in Bd(A)$ then there exists a set $X in x$ that intersects both $X$ and $X-A$ since $A subset Cl(A)$ and $X-A subseteq Cl(X-A)$
How do I finish this? Am I on the right track?
Let $x in Cl(A) cap Cl(X-A)$ then $x in Cl(A)$ and $x in Cl(X-A)$. Where do I go from here?