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## Menger universal continua

The Menger universal continuum , for [Menger 1926], can be defined as follows [Engelking 1978, pp. 121-122].

Let . Inductively, suppose a collection of cubes has been defined for . Divide each cube into congruent cubes with edges of length . If is the collection of all these cubes that intersect -dimensional faces of , then let .

Define

The most popular continua are , widely known as the Sierpinski universal plane curve or the Sierpinski carpet [Sierpinski 1916], and , called the Menger universal curve [Menger 1926]. These particular curves are discussed separately.

Continuum can also be obtained as the inverse limit in the following way. Let be an open base of consisting of open -cells such that . Put and , where relation identifies points iff . The bonding maps are natural retractions [Pasynkov 1965], [Bestvina 1988, pp. 98-99].

1. is the universal space in the class of all compacta of dimension which embed in [Štanko 1971] ([Menger 1926] for ). is universal in the class of all -dimensional subsets of ([Engelking 1978, Problem 1.11.D]), and in the class of all -dimensional separable metric spaces [Bothe 1963].
2. A continuum is homeomorphic to if and only if can be embedded in the -sphere in such a way that has infinitely many components such that , for , is an -cell for each and is dense in (see [Cannon 1973] and [Chigogidze et al. 1995, Theorem 6.1.2, p. 74]).
3. A continuum is homeomorphic to if and only if is -dimensional, and -space with the disjoint -disks property ( property) [Bestvina 1988, Corollary 5.2.3, p.98].

Equivalently, is homeomorphic to if and only if is -dimensional, and -space such that any continuous map from a compact space of dimension into is a uniform limit of embeddings (Z-embeddings) [Bestvina 1988, Theorem 2.3.8, p. 36].

1. If is a Z-setZ-set, then, for every open neighborhood of and every , there is such that if is a Z-set and is a -homeomorphism, then extends to an -homeomorphism such that [Bestvina 1988, Theorem 3.1.1, p. 65].
2. For every , there exists such that if are Z-setsZ-set in and is a -homeomorphism, then there is an -homeomorphism extending [Bestvina 1988, Theorem 3.1.3, p. 71].

3. Every homeomorphism between Z-sets in extends to an autohomeomorphism of [Bestvina 1988, Corollary 3.1.5, p. 72].

It follows that is strongly locally homogeneous and countably dense homogeneous.

4. If , then is not homogeneous [Lewis 1987] (for the homogeneity of follows from the property above).

5. is not 2-homogeneous for an arbitrary continuum [Kuperberg et al. 1995].

6. The groups of autohomeomorphisms of and are Polish and one-dimensional (see [Oversteegen et al. 1994, Corollary 6] and Property 2).

The group is totally disconnected for (see [Brechner 1966, Theorem 1.3] for and its natural extension for any ).

7. The Polish group of autohomeomorphisms of is at most one-dimensional [Oversteegen et al. 1994].

8. The group of autohomeomorphisms of is simple (see [Chigogidze et al. 1995, Theorem 3.2.4]).

9. Every autohomeomorphism of is a composition of two homeomorphisms, each of which is the identity on some nonempty open set [Sakai 1994].

10. For each , there is a copy of with the Hausdorff dimension [Chigogidze et al. 1995, Theorem 4.2.6].

11. Every Z-set in is the fixed point set of some autohomeomorphisms of [Sakai 1997].

12. Any and -continuum is the image of under a -map [Bestvina 1988, Theorem 5.1.8, p. 95].

13. If a Polish space contains a topological copy of or , then the space of all copies of in with the Hausdorff metric is not a -set [Krupski XXXXb].

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Next: Sierpinski carpet Up: Cyclic examples of locally Previous: Cyclic examples of locally
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30