I'm really stuck on this Real Analysis problem, if anyone would mind helping me.
Let $(X,d)$ be a metric space, and let $A$ be a non-empty subset of $X$.
How do you show whether any point $x \in X$ is in the interior, interior of the complement, or boundary of $A$, if the only information you have is $d_A(x) = \inf d(x,a)$ and $d_{A^c}(x)$.
I have been trying to make some arguments using strictly distances but this hasn't been working out. How would you do this?
Thanks.