{Menger universal curve}
\label{ex-Menger curve}

The
\e{Menger universal curve}
$M^3_1$ is the subset of the unit cube $I^3$ whose
projections onto faces of the cube are the
\gs{Sierpi\'nski carpets}{Sierpinski carpet}, i.e.,
$$M^3_1=\{\,(x,y,z)\in I^3: (x,y)\in M^2_1,\quad (y,z)\in M^2_1,
\quad (x,z)\in M^2_1\,\}$$  \cite{Menger1926a}.
Other descriptions can be found in  \ref{ex-Menger}
(see also \cite[pp. 5--6, 8--9]{Mayer+1986a}).
See Figure A.

\FIGURE{A}{Menger universal curve}

\GIF{AA}{Menger universal curve - an animation}
%insert Figure 3.3.3

\rostere

\item
The following statements are equivalent:
\rostere
\item
$X$ is homeomorphic to $M^3_1$;
\item
$X$ is a locally connected curve with no local cut points
and no planar open nonempty subsets \cite[Theorem XII, p. 13]{Anderson1958a};
\item
$X$ is a \g{homogeneous} locally connected \g{curve}
different from a \g{simple close curve} \cite[Theorem XIII, p.
14]{Anderson1958a};
\item
$X$ is a locally connected curve with the \g{disjoint arcs property} (see
Property~\ref{ex-Menger-Menger3} in \ref{ex-Menger});
\item
$X$ is a locally connected curve and each arc in $X$ is
\g{approximately non-locally-separating arc} and has empty interior in $X$
\cite[Theorem 3, p. 86]{Krupski+XXXXa}.
\rosteree

\item
$M^3_1$ is \gs{universal}{universal space} in the class of all
metric separable spaces of dimension $\le 1$
\cite{Menger1926a}.

\item
 Z-sets\g{Z-set} in $M^3_1$ coincide with
 \gs{non-locally-separating closed subsets}{non-locally-separating set}  of $M^3_1$.

If $Z$ is a separable metric space of dimension $\le 1$,
then $Z$ can be embedded as a non-locally-separating subset of $M^3_1$
\cite[Theorem 6.1, p. 42]{Mayer+1986a}.

\item
If $Z$ is a closed subset of a metric space $X$,
$\dim(X\setminus Z)\le 1$ and $f:Z\to M^3_1$ is a
continuous mapping with non-locally separating image $f(Z)$,
then $f$ can be extended to a map $g:X\to M^3_1$
such that $g|X\setminus Z$ is an embedding into
$M^3_1\setminus f(Z)$ \cite[Theorem 6.4, p. 44]{Mayer+1986a}.

\item
Every continuous surjection $h$ between
non-locally separating closed subsets $Z,Z'$  of $ M^3_1$
extends to a mapping $h^*$ of $M^3_1$ such that
$h^*|M^3_1\setminus Z$ is a homeomorphism onto
$M^3_1\setminus Z'$ \cite[Corollary 6.5, p. 44]{Mayer+1986a};
moreover, for every $\epsilon>0$ there exists $\delta>0$
such that if $h$ is a $\delta$-homeomorphism,
then $h^*$ can be taken as an $\epsilon$-homeomorphism
(see Property~\ref{ex-Menger-Menger4b} in \ref{ex-Menger}).

\item
For each locally connected continuum $X$, there exist open surjections
$f:M^3_1\to X$ and $g:M^3_1\to X$  such that $f^{-1}(x)$ is
homeomorphic to  $M^3_1$ and $g^{-1}(x)$ is a Cantor set,
for any $x\in X$ \cite{Wilson1972a}.

\item
Any compact 0-dimensional group $G$ acts freely on $M^3_1$
so that the orbit space $M^3_1/G$ is homeomorphic to $M^3_1$
\cite[Theorem 1]{Anderson1957a}.

\item
If a locally compact space $X$ contains a topological copy
of $M^3_1$, then the space of all  copies of $M^3_1$ in  $X$
with the Hausdorff metric\g{Hausdorff metric} is a true
absolute $F_{\sigma\delta}$-set
\cite{KrupskiXXXXb}.

\rosteree
