The Freudenthal compactification of tree-like generalized continua (Q2850649)

A generalized continuum is a locally compact connected metric space. A continuous map \(f: X \rightarrow Y\) is said to be proper if for any compact subset \(K \subset Y\), \(f^{-1}(K)\) is compact in \(X\). A generalized continuum is tree-like if it can be expressed as the inverse limit of an inverse sequence of locally compact trees with proper bonding maps.NEWLINENEWLINEIn this paper it is shown that the following statements are equivalent: (1) the generalized continuum \(X\) is tree-like; (2) the Freundenthal compactification \(\hat{X}\) is tree-like. Next the result is used to prove that the tree-likeness of generalized continua is preserved by end-faithful proper confluent surjections, which is an affirmative answer to the question posed by the second and third author in [Topology Appl. 156, No. 18, 2960--2970 (2009; Zbl 1254.54025)].