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In connection with the characterization of dendrites having
closed set of their end points (see
Property 2 in 1.3.4 above)
the classes of those dendrites that do not contain a copy of
and those that do not contain a copy of the
comb are
of some interest (see [Arévalo et al. 2001, p. 8]). The first
one is just the class of dendrites with finite orders of
ramification points, and it is known to have a universal
element according to Property 4
in 1.3.8. The other class has
also a universal element . Its construction, given in
[Arévalo et al. 2001, p. 9] is the following.
Let be any point in the plane
and let
be
a sequence of points in
tending to and such that no three
of the points
are collinear. Define
. Then is homeomorphic to
. For every
, let be a point in
such that for any fixed the sequence
tends to , no three of the points
are collinear, and that the continuum
is a dendrite. We continue
constructing dendrites
in the same manner, such that we
get an increasing sequence of dendrites
in
having the property that the closure of their union is also a dendrite.
Finally define
The following properties of the
dendrite
are proved in [Arévalo et al. 2001, Theorems 4.5 and 4.6, p.
10].
 A dendrite is homeomorphic to the dendrite if and only if
each of the following conditions is satisfied:
 contains no copy of ;
 the set of all end points of is contained in the closure of
the set of all ramification points of ;
 all ramification points of are of order .
 The dendrite is universal in the class
of all dendrites containing no copy of .
See Figure A.
Figure 1.3.9:
( A ) dendrite

Figure:
( AA ) dendrite  an animation

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: Selfhomeomorphic dendrites
Up: Dendrites
Previous: Other universal dendrites
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
20011130