1
$\begingroup$

Let $X$ be topological space and $A\subset X$. Now, $X$ has a discrete topology. Determine $\partial A$, the boundary.

Is $\partial A$ empty set?

$\endgroup$
2
$\begingroup$

Sure. If you know the definition of $\partial A = \overline{A}\setminus A^{\circ}$, then this is clear, as for all subsets $A$ we have $A = A^{\circ}$ (all sets are open) and $A = \overline{A}$ (all sets are closed).

Also, if $A \subset X$ and $x \in X$, the neighbourhood $\{x\}$ of $x$ cannot intersect both $A$ and $X \setminus A$, whatever $A$ is, so $x \notin \partial A$ (this is another equivalent definition of boundary).

$\endgroup$

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.