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.

- is universal in the class of all at most one-dimensional subsets of the plane (equivalently, of all boundary subsets of the plane) [Sierpinski 1916], [Sierpinski 1922].
- The following statements are equivalent:
- is homeomorphic to ;
- is a locally connected plane curve that contains no local cut points;
- is a continuum embeddable in the plane in such a way that has infinitely many components such that , for , is a simple closed curve for each and is dense in [Whyburn 1958].

- A complete metric space contains a topological copy of if and only if contains a subset with the bypass property [Prajs 1998a].
- The group of all autohomeomorphisms of has exactly
two orbits: one of them is the union of all simple
closed curves which are the
boundaries of complementary domains of
[Krasinkiewicz 1969].
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).

- Any homeomorphism between Cartesian products of copies of
the Sierpinski carpet is factor preserving [Kennedy Phelps 1980]. Consequently,
no such product is homogeneous.
- The Sierpinski carpet can be continuously
decomposed into
pseudo-arcs such that the decomposition
space is homeomorphic to the carpet [Prajs 1998b, Corollary
18], [Seaquist 1995]. In fact, is
the only planar locally connected curve admitting such a
decomposition [Prajs 1998b, Corollary 18].
- The Sierpinski carpet is
homogeneous with respect to
monotone open mappings
[Prajs 1998b, Corollary 24], [Seaquist 1999].
Moreover, every continuum which is locally homeomorphic to
(i. e., -manifold) is homogeneous with respect to
monotone open mappings
[Prajs 1998b, Theorem 23].
- is homogeneous with respect to the class of
simple mappings
[Charatonik 1984].
- If is a curve, then the set of all mappings
such that is homeomorphic to is a
residual subset of the space
of all mappings of into
with the uniform convergence metric.
If is the hyperspace of all subcontinua of a compact space and its subspace of all curves, then the set

$f(C)$ is homeomorphic to $M^2_1$is residual in ; in other words, almost all mappings in map almost all curves in onto copies of , where "almost all" means all with except of a subset of the first category in corresponding spaces [Mazurkiewicz 1938]. - If a locally compact space contains a topological copy
of the Sierpinski carpet, then the space of all copies of
the Sierpinski carpets in with the Hausdorff
metricHausdorff metric is a true absolute
-set [Krupski XXXXb].