1. ## open/closed sets

Well I guess this is not standard notation but here goes:

Let $A$ be a set in $\mathbb{R}^n$, then $\partial A$ is the boundry of A.
$\overline{A}=A\cup\partial A$
and
$\overset{\circ}{A}=A\setminus\partial A$

Now prove that
$\overset{\circ}{\overset{\_}{\overset{\circ}{\over set{\_}{A}}}}=\overset{\circ}{\overset{\_}{A}}$

and

$\overset{\_}{\overset{\circ}{\overset{\_}{\overset {\circ}{A}}}}=\overset{\_}{\overset{\circ}{A}}$

I just have no idea how to do this one. Any hints greatly appriciated.

2. Originally Posted by hjortur
Well I guess this is not standard notation but here goes:

Let $A$ be a set in $\mathbb{R}^n$, then $\partial A$ is the boundry of A.
$\overline{A}=A\cup\partial A$
and
$\overset{\circ}{A}=A\setminus\partial A$

Now prove that
$\overset{\circ}{\overset{\_}{\overset{\circ}{\over set{\_}{A}}}}=\overset{\circ}{\overset{\_}{A}}$

and

$\overset{\_}{\overset{\circ}{\overset{\_}{\overset {\circ}{A}}}}=\overset{\_}{\overset{\circ}{A}}$

I just have no idea how to do this one. Any hints greatly appriciated.
I would just honestly try it. It isn't as ugly as it may appear. Let $x$ be in the LHS and deduce that it's in the RHS. Come back if you need help.