Sierpiski Universal Plane Curve

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 $\cdots$ . 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].

Sierpi\' nski Universal Plane Curve- fig. a


Sierpi\' nski Universal Plane Curve- fig. b


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