# Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$

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 $$∂A = \text{Cl}(A) − \text{Int}(A)$$"

• Use the fact that set subtraction is the same as intersecting the complement. – Tony S.F. Jun 16 at 18:12

$$\partial A= \mathrm{Cl}(A)-\mathrm{Int}(A)=\mathrm{Cl}(A) \cap (X - \mathrm{Int}(A))$$. 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.
Since, $$\partial A=\overline A\setminus\mathring A$$, $$(\partial A)^\complement=\overline A^\complement\cup\mathring A$$, which is open, since it's the union of two open sets.