The Sierpinski universal plane curve or the Sierpinski carpet [Sierpinski 1916] is a well known fractal obtained as the set remaining when one begins with the unit square and applies the operation of dividing it into 9 congruent squares and deleting the interior of the central one, then repeats this operation on each of the surviving 8 squares, and so on. See Figure A.
The group is a Polish topological group which is totally disconnected and one-dimensional (see [Brechner 1966, Theorem 1.2] and Property 7 in 1.4.1).
If is the hyperspace of all subcontinua of a compact space and its subspace of all curves, then the set