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like

If X$ is a metric space, a mapping f$ from X$ to a space Y$ is an \varepsilon$-map if, for each point y$ of Y$, diamf^{-1}(y)) \leq \varepsilon$. If C$ is a collection of continua, a continuum M$ is C$-like if, for every positive number \varepsilon$, there exists an \varepsilon$-map of M$ onto an element of C$. In particular, a continuum is tree-like if, for some collection C$ of trees, M$ is C$-like. A concept of a tree-like continuum can be defined in several (equivalent) ways. One of them is the following. A continuum X$ is said to be tree-like provided that for each \varepsilon > 0$ there is a tree T$ and a surjective mapping f: X \to T$ such that f$ is an \varepsilon$-mapping (i.e., diam^{-1}(y) < \varepsilon $ for each y \in T$). Let us mention that a continuum X$ 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 X$ is be tree-like provided that for each \varepsilon > 0$ there is an \varepsilon$-covering of X$ 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 up previous contents index
Next: locally Up: Definitions Previous: light
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30