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## 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].

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Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30