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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 $∂A = \text{Cl}(A) − \text{Int}(A)$"

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    $\begingroup$ Use the fact that set subtraction is the same as intersecting the complement. $\endgroup$ – Tony S.F. Jun 16 at 18:12
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$\partial A= \mathrm{Cl}(A)-\mathrm{Int}(A)=\mathrm{Cl}(A) \cap (X - \mathrm{Int}(A))$. 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.

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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.

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