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The Menger universal continuum
, for [Menger 1926], can
be defined as follows [Engelking 1978, pp. 121122].
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. 9899].
 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].
 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]).
 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 (Zembeddings)
[Bestvina 1988, Theorem 2.3.8, p. 36].
 If
is a ZsetZset, then, for
every open neighborhood
of and
every
, there is such that if
is a Zset and is a
homeomorphism, then extends to an
homeomorphism
such that
[Bestvina 1988, Theorem 3.1.1, p. 65].
 For every
, there exists
such that if are ZsetsZset in
and is a homeomorphism,
then there is an homeomorphism
extending [Bestvina 1988, Theorem 3.1.3, p. 71].
 Every homeomorphism between Zsets 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.
 If , then is not homogeneous
[Lewis 1987] (for the homogeneity of
follows from the property above).

is not 2homogeneous for an arbitrary
continuum [Kuperberg et al. 1995].
 The groups of autohomeomorphisms of
and
are Polish and onedimensional
(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 ).
 The Polish group of autohomeomorphisms of is
at most onedimensional [Oversteegen et al. 1994].
 The group of autohomeomorphisms of
is
simple (see [Chigogidze et al. 1995, Theorem
3.2.4]).
 Every autohomeomorphism of
is
a composition of two homeomorphisms,
each of which is the
identity on some nonempty open set [Sakai 1994].
 For each
, there is a copy of
with the
Hausdorff dimension [Chigogidze et al. 1995, Theorem
4.2.6].
 Every Zset in
is the fixed point set of
some autohomeomorphisms of
[Sakai 1997].
 Any and continuum is the image of
under a map
[Bestvina 1988, Theorem 5.1.8, p. 95].
 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].
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: Sierpinski carpet
Up: Cyclic examples of locally
Previous: Cyclic examples of locally
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
20011130