real analysis - Metric Spaces, determining whether a point is in the interior, boundary, or interior of the complement.

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.

Answer

$\newcommand{\cl}{\operatorname{cl}}\newcommand{\int}{\operatorname{int}}\newcommand{\bdry}{\operatorname{bdry}}$I gather from your comment to Berci that you have the basic idea but are having a hard time expressing it. It might help to start by proving the observation that for any non-empty $S\subseteq X$ and any $x\in X$, $d_S(x)=0$ if and only if $x\in\cl S$. Then you immediately get that

$$d_A(x)=0=d_{X\setminus A}(x)\quad\text{iff}\quad x\in\cl A\cap\cl(X\setminus A)=\bdry A\;,$$

and you should find it even easier to write down and verify the conditions on $d_A(x)$ and $d_{X\setminus A}(x)$ corresponding to $x\in\int A$ and $x\in\int(X\setminus A)$.