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