If is a metric space, a mapping from to a space
-map if, for each point of ,
. If is a
collection of continua, a continuum is
-like if, for every positive number
-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
such that is an
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].
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih