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For a given nonempty set
we denote by
any dendrite
satisfying the following two conditions:
- (a)
- if
is a ramification point of
, then
;
- (b)
- for each arc
contained in
and for every
there is in
a point
with
.
It is shown in [Charatonik W.J. et al. 1994, Theorem 6.2, p. 229]
that the
dendrite
is topologically unique, i.e., if
two dendrites satisfy conditions (a) and (b) with the same
set
, then they are
homeomorphic. The dendrite
is called the
standard universal dendrite of orders in
. If
is a singleton
,
then
is just the
standard universal dendrite
defined previously.
The following properties of dendrites
are known (see [Charatonik W.J. et al. 1994, Theorems
6.4 and 6.6-6.8, p. 230; Corollary 6.10, p. 232]).
- For any nonempty subset
, the dendrite
is strongly pointwise self-homeomorphic.
- If
, then the dendrite
is
universal
for the family of all dendrites.
- If the set
is finite with
, then
is
universal for the family of all dendrites having orders of ramification
points at most
.
- If the set
is infinite and
,
then
is universal for the family of all dendrites having finite orders
of ramification points.
- Nonconstant open images of standard universal
dendrites
are homeomorphic to
if and only if
is a nonempty
subset of
.
- For any nonempty subset
and
for an arbitrary dendrite
there exists a monotone mapping from
onto
,
[Charatonik et al. 1998, Theorem 2.22, p. 239].
- For any nonempty subset
the dendrite
is monotonely homogeneous, [Charatonik 1996, Theorem 3.3, p. 292].
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: The dendrite
Up: Dendrites
Previous: Universal dendrites of order
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30