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. [65, 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 [4, p. 653].