Next: 2-dimensional locally connected continua Up: Cyclic examples of locally Previous: Sierpinski carpet

## Menger universal curve

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.

1. The following statements are equivalent:
1. is homeomorphic to ;
2. is a locally connected curve with no local cut points and no planar open nonempty subsets [Anderson 1958, Theorem XII, p. 13];
3. is a homogeneous locally connected curve different from a simple close curve [Anderson 1958, Theorem XIII, p. 14];
4. is a locally connected curve with the disjoint arcs property (see Property 3 in 1.4.1);
5. 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].

2. is universal in the class of all metric separable spaces of dimension [Menger 1926].

3. 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].

4. 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].

5. 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).

6. For each locally connected continuum , there exist open surjections and such that is homeomorphic to and is a Cantor set, for any [Wilson 1972].

7. Any compact 0-dimensional group acts freely on so that the orbit space is homeomorphic to [Anderson 1957, Theorem 1].

8. 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