BACKGROUND MATERIAL

The open ball $B(x;r) : = \left\{ y \in \Re^n : \vert\vert x-y\vert\vert < r \right\}.$ Suppose that $D$ is a subset of $\Re^n$.

Interior:

$x \in {\rm int\,}D$ if there exists $r > 0$ with $B(x;r) \subset D$.

Closure:

$x \in {\rm cl\,}D$ if there exists a sequence $x^k \in D$ with $x^k \rightarrow x.$

Boundary:

$x \in \partial D$ if $x\in {\rm cl\,}D \backslash {\rm int\,}D.$

$D$ is open if $D = {\rm int\,}D.$ $D$ is closed if $D = {\rm cl\,}D.$