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The following properties of dendrites are known (see e.g. [Nadler 1992, Chapter 10,
p. 165], [Kuratowski 1968, §51, VI, p. 300]
or [Whyburn 1942, p. 88]; compare
also [Charatonik et al. 1998, Theorems 1.3 and 1.5, p. 233]).
- Every subcontinuum of a dendrite is a dendrite (thus each dendrite is
hereditarily locally connected).
- Every connected subset of a dendrite is arcwise connected.
- In any dendrite the order of a point is equal to the number of
components of
.
- The set of all end points of a dendrite is 0-dimensional.
- The set of all end points of a dendrite is a -set.
- The set of all ordinary points (i.e., points of
order 2) of a dendrite is dense in .
- Every dendrite has at most countably many ramification points.
- If is the order of a ramification point in a dendrite, then
.
- No dendrite contains points of order or
(i.e., each dendrite is a regular continuum).
- In any dendrite the set of all its end points is dense if and only if
the set of all its ramification points is dense.
- Dendrites are absolute retracts. In fact, the
class of all 1-dimensional absolute retracts coincide with the class of
dendrites, [Borsuk 1967, Corollary 13.5, p. 138].
- A continuum is a dendrite if and only if for every compact space
(continuum) and for every light
confluent mapping
such that
there
is a copy of in for which the restriction
is a
homeomorphism, [Charatonik et al. 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: Simple examples
Up: Dendrites
Previous: Dendrites
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30