Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that $∂A = \text{Cl}(A) − \text{Int}(A)$. If that were the case, I could just state that definition and note that it's the intersection of closed sets.
I have very little knowledge of set theory and proofs, so I'm not sure how else to prove this. As always, I appreciate any help.
The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa.
"Let $A$ be a subset of a topological space $X$. Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$"
Best Answer
$\partial A= \mathrm{Cl}(A)-\mathrm{Int}(A)=\mathrm{Cl}(A) \cap (X - \mathrm{Int}(A))$. Now $\mathrm{Cl}(A)$ is closed by definition, $X-\mathrm{Int}(A)$ is closed since it's the complement of an open, and intersection of closed is closed, so $\partial A$ is closed.