Prove that a point belongs to $A^{-}$ if and only if it is either an interior or a boundary point of $A$; where $A^-$ is the closure.
To be an interior point: A point $x\in \mathbb{R}$ is the interior point of a set $A\subset \mathbb{R}$ if there is a neighborhood of $x$ which is entirely contained in $A$.
A boundary point follows, which is the set of points with the property that every open set containing the point intersects the interior of $A$ and the interior of $A^c$. To be a closure, which is the intersection of all closed sets containing $A$; to be closed means there is a boundary point. But, how will it follow that it can also be an interior point?
Best Answer
A boundary point of a set $A$ cannot be an interior point, but that’s not a problem: the theorem is that the closure of $A$ consists of those points that are in the interior of $A$ together with those that are in the boundary of $A$. In symbols,
$$\operatorname{cl}A=\operatorname{int}A\cup\operatorname{bdry}A\;,\tag{1}$$
and in fact the two sets on the righthand side of $(1)$ are disjoint. (By the way, a closed set need not have any boundary points at all: in $\Bbb R$ the only examples of this phenomenon are the closed sets $\varnothing$ and $\Bbb R$, but in more general topological spaces there can be many sets that are simultaneously open and closed and which therefore have empty boundary.)
You know that $x\in\operatorname{int}A$ if and only if $x$ has an open nbhd contained in $A$, and that $x\in\operatorname{bdry}A$ if and only if every open nbhd of $x$ intersects both $A$ and $\Bbb R\setminus A$. To prove $(1)$, I suggest that you first prove that
Once you have this, $(1)$ is pretty straightforward.