The
*Menger universal curve*
is the subset of the unit cube whose
projections onto faces of the cube are the
Sierpinski carpets, i.e.,

- The following statements are equivalent:
- is homeomorphic to ;
- is a locally connected curve with no local cut points and no planar open nonempty subsets [Anderson 1958, Theorem XII, p. 13];
- is a homogeneous locally connected curve different from a simple close curve [Anderson 1958, Theorem XIII, p. 14];
- is a locally connected curve with the disjoint arcs property (see Property 3 in 1.4.1);
- is a locally connected curve and each arc in is approximately non-locally-separating arc and has empty interior in [Krupski et al. XXXXa, Theorem 3, p. 86].

- is universal in the class of all
metric separable spaces of dimension
[Menger 1926].
- Z-setsZ-set in coincide with
non-locally-separating closed subsets of .
If is a separable metric space of dimension , then can be embedded as a non-locally-separating subset of [Mayer et al. 1986, Theorem 6.1, p. 42].

- If is a closed subset of a metric space ,
and
is a
continuous mapping with non-locally separating image ,
then can be extended to a map
such that
is an embedding into
[Mayer et al. 1986, Theorem 6.4, p. 44].
- Every continuous surjection between
non-locally separating closed subsets of
extends to a mapping of such that
is a homeomorphism onto
[Mayer et al. 1986, Corollary 6.5, p. 44];
moreover, for every
there exists
such that if is a -homeomorphism,
then can be taken as an -homeomorphism
(see Property 4b in 1.4.1).
- For each locally connected continuum , there exist open surjections
and
such that is
homeomorphic to and is a Cantor set,
for any [Wilson 1972].
- Any compact 0-dimensional group acts freely on
so that the orbit space is homeomorphic to
[Anderson 1957, Theorem 1].
- If a locally compact space contains a topological copy
of , then the space of all copies of in
with the Hausdorff metricHausdorff metric is a true
absolute
-set
[Krupski XXXXb].