A and A complement have the same boundary

[Ted123]

How would you show that if $X$ is a non-empty set and $A\subseteq X$ then $A$ and $A^c$ have the same boundary?

The definition is $x\in \partial A \iff$ there exists $r>0$ such that the open ball $B(x,r)$ intersects both $A$ and $A^c$

but this is precisely the statement that $x\in \partial A^c$!

 

[micromass]

How would you show that if $X$ is a non-empty set and $A\subseteq X$ then $A$ and $A^c$ have the same boundary?

The definition is $x\in \partial A \iff$ there exists $r>0$ such that the open ball $B(x,r)$ intersects both $A$ and $A^c$

but this is precisely the statement that $x\in \partial A^c$!

Indeed, so the proof is trivial.