
The
{{{\it Sierpi\' nski Universal Plane Curve}
\label{Sierpinski Universal Plane Curve}
\index{Sierpinski Universal Plane Curve@Sierpi\' nski Universal Plane Curve}}}
is a well known \g{continuum} which serves as the universal
element in the class of all
\g{one-dimensional} continua in the plane. It is obtained as the
residual set remaining when one begins with a square and
applies the operation of dividing it  into nine equal
squares and omitting the interior of the center one, then
repeats this operation on each of the surviving 8 squares,
then repeats again on the surviving 64 squares, and so on
$\cdots$ . See \cite[p.9]{nadler1992}.

The Sierpi\' nski Universal Plane Curve can be characterized
as the only plane \g{locally connected} \g{one-dimensional}
\g{continuum} $S$ such that the boundary of each complementary
domain of $S$ is a \g{simple closed curve} and no two of
these complementary domain boundaries intersect. See
\cite{whyburn1958}.
