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The
* Sierpiski Universal Plane Curve*
is a well known continuum which serves as the universal
element in the class of all
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
. See [65, p.9].
The Sierpiski Universal Plane Curve can be characterized
as the only plane locally connected one-dimensional
continuum *S* such that the boundary of each complementary
domain of *S* is a simple closed curve and no two of
these complementary domain boundaries intersect. See
[79].

Figure (
A
)
Sierpiski Universal Plane Curve

Figure (
B
)
Sierpiski Universal Plane Curve

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*Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih*

*2001-02-21*