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The
Menger universal curve
is the subset of the unit cube whose
projections onto faces of the cube are the
Sierpinski carpets, i.e.,
[Menger 1926].
Other descriptions can be found in 1.4.1
(see also [Mayer et al. 1986, pp. 5-6, 8-9]).
See Figure A.
Figure 1.4.3:
( A ) Menger universal curve
|
Figure:
( AA ) Menger universal curve - an animation
|
- 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].
Here you can find source files
of this example.
Here you can check the table
of properties of individual continua.
Here you can read Notes
or
write to Notes
ies of individual continua.
Next: 2-dimensional locally connected continua
Up: Cyclic examples of locally
Previous: Sierpinski carpet
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30