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 $∂A = \text{Cl}(A) − \text{Int}(A)$"

  • 1
    $\begingroup$ Use the fact that set subtraction is the same as intersecting the complement. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.