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If
is a metric space, a mapping
from
to a space
is an
-map if, for each point
of
,
. If
is a
collection of continua, a continuum
is
-like if, for every positive number
, there
exists an
-map of
onto an element of
.
In particular, a continuum is tree-like if, for some
collection
of trees,
is
-like.
A concept of a tree-like continuum can be defined in several
(equivalent) ways. One of them is the following. A continuum
is said to be tree-like
provided that for each
there is a tree
and a surjective
mapping
such that
is an
-mapping (i.e.,
for each
). Let us mention that a continuum
is
tree-like if and only if it is the inverse limit of an inverse
sequence of trees with surjective bonding mappings. Compare e.g.
[Nadler 1992, p. 24].
Using a concept of a nerve of a covering, one can reformulate the above
definition saying that a continuum
is be tree-like provided that for each
there is an
-covering of
whose nerve is a tree.
Finally, the original definition using tree-chains can be found e.g. in
Bing's paper [Bing 1951, p. 653].
Next: locally
Up: Definitions
Previous: light
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30