Finding the true statements about the boundary of a set Pick out the true statements:
(a) If $A$ is open, then $∂A$ is nowhere dense.
(b) If $A$ is closed, then $∂A$ is nowhere dense.
(c) If $A$ is any subset, then $∂A$ is always nowhere dense.     
$∂A$  represents the boundary of a set

I am stuck on this problem and seeking your help.
 A: As you might be aware, for any set $A$, the relation $\partial A = \partial (A^c)$ holds. Here, $A^c$ is the set-theoretic complement of $A$. Since the open sets and the closed sets are complements of one another, you should be able to see that there is a close relationship between statements (a) and (b). In fact, they are both equivalent. It is enough to decide if one of them is true/false, as the answer for the other will be the same.
Some more points you may be aware of:


*

*$\partial A$ is always closed, for any set $A$. 

*To decide if a closed set, such as $\partial A$, is nowhere dense, you just need to decide whether it has an interior point. 

*Another way of saying that a set $U$ is open is to say that it contains none of its boundary points, i.e. $U \cap \partial U = \varnothing$.


If you want to showthat an open set $U$ has nowhere dense boundary $\partial U$, then you should show that each point $p \in \partial U$ is not an interior point. Indeed, any neighbourhood of $p$ will meet $U$ (since it is a boundary point) and points of $U$ are not in $\partial U$ (see above), so $\partial U$ is nowhere dense. This answers (a), hence answers (b) as well.
(c) is false. I will leave you with the following hint: try to come up with an example of a set $A$ such that $\partial A$ is actually the whole space (hence definitely not nowhere dense).
