{Sierpi\'nski carpet}

The
\es{Sierpi\'nski universal plane curve}{Sierpinski universal plane curve}
or the
\es{Sierpi\'nski carpet}{Sierpinski carpet}
\cite{Sierpinski1916a} $M^2_1$ is a well known \g{fractal}
obtained as the set remaining when one begins with the unit
square $I^2$ 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.

\FIGURE{A}{Sierpi\'nski carpet}
%Insert Figure 3.3.2

\GIF{AA}{Sierpi\'nski carpet - an animation }

\rostere
\item
$M^2_1$ is \g{universal} in the class of all at most
one-dimensional subsets of the plane (equivalently, of all
boundary subsets of the plane) \cite{Sierpinski1916a},
\cite{Sierpinski1922a}.
\item
The following statements are equivalent:
\rostere
\item
 $X$ is homeomorphic to $M^2_1$;
\item
$X$ is a \g{locally connected} plane \g{curve} that contains
no \gs{local cut points}{local cut point};
\item
$X$ is a continuum embeddable  in the plane in such a way
that $\mathbb R^2\setminus  X$ has infinitely  many
components $C_1, C_2,\dots$ such that $\diam C_i\to 0$, $\bd
C_i\cap \bd C_j=\emptyset$ for $i\ne j$, $\bd C_i$ is a
\g{simple closed curve} for each $i$ and $\bigcup_{i=1}^\infty
\bd C_i$ is dense in $X$ \cite{Whyburn1958a}.
\rosteree
\item
A complete metric space $X$ contains a topological copy of
$M^2_1$ if and only if $X$ contains a subset with the
\g{bypass property} \cite{Prajs1998a}.
\item
The group of all autohomeomorphisms of $M^2_1$ has exactly
two orbits: one of them is the union of  all \gs{simple
closed curves}{simple closed curve} which are the
boundaries of complementary domains of $M^2_1$
\cite{Krasinkiewicz1969a}.

The group is a Polish topological group which is totally
disconnected and one-dimensional (see \cite[Theorem
1.2]{Brechner1966a} and Property~\ref{ex-Menger-autohomeo}
in \ref{ex-Menger}).

\item
Any homeomorphism between Cartesian products of copies of
the Sierpi\'nski carpet is  \gs{factor preserving}{factor
preserving homeomorphism} \cite{Kennedy1980a}. Consequently,
no such product is \g{homogeneous}.

\item
The Sierpi\'nski carpet can be \gs{continuously
decomposed}{continuous decomposition} into
\gs{pseudo-arcs}{pseudo-arc} such that the decomposition
space is homeomorphic to the carpet \cite[Corollary
18]{Prajs1998b}, \cite{Seaquist1995a}. In fact, $M^2_1$ is
the only planar locally connected curve admitting such a
decomposition \cite[Corollary 18]{Prajs1998b}.

\item
The Sierpi\'nski carpet is
\gs{homogeneous with respect to}
{homogeneous with respect to a class of mappings}
\gs{monotone open mappings}{monotone open mapping}
\cite[Corollary 24]{Prajs1998b}, \cite{Seaquist1999a}.
Moreover, every continuum which is locally homeomorphic to $M^2_1$
(i. e., $M^2_1$-manifold) is homogeneous with respect to
monotone open mappings
\cite[Theorem 23]{Prajs1998b}.

\item
$M^2_1$ is homogeneous with respect to the class of
\gs{simple mappings}{simple mapping}
\cite{Charatonik1984a}.

\item
If $C$ is a curve, then the set of all mappings $f:C\to \mathbb R^2$
such that $f(C)$ is homeomorphic to $M^2_1$ is a
\gs{residual}{residual set} subset of the space $(\mathbb R^2)^C$
of all mappings of $C$ into $\mathbb R^2$
with the uniform convergence metric.

If $C(X)$ is the \gs{hyperspace}{hyperspace of subcontinua}
of all subcontinua of a compact
space $X$ and $C_1(X)$ its subspace of all curves, then the set
$$\{\,(C,f)\in C(X)\times (\mathbb R^2)^X: \text{$f(C)$
is homeomorphic to $M^2_1$}\,\}$$
is residual in  $C(X)\times (\mathbb R^2)^X$;
in other words, almost all mappings in $(\mathbb R^2)^X$ map
almost all curves in $X$ onto copies of $M^2_1$,
where "almost all" means all with except of a subset of the
first category in corresponding spaces \cite{Mazurkiewicz1938a}.

\item
If a locally compact space $X$ contains a topological copy
of the Sierpi\'nski carpet, then the space of all  copies of
the Sierpi\'nski  carpets in  $X$ with the Hausdorff
metric\g{Hausdorff metric} is a true absolute
$F_{\sigma\delta}$-set \cite{KrupskiXXXXb}.

\rosteree
