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**Lines, trees, and branch spaces.**
*(English)*
Zbl 1025.54016

It is well known that starting with a linearly ordered topological space LOTS (in our context such a space will be called a line), one can construct a tree whose members are convex subsets of the original line and that sometimes reflects crucial properties of the line. Alternatively, one can start with an abstract tree (not necessarily coming from convex sets in a given linearly ordered set) and construct its branch spaces, often obtaining lines with interesting properties. The authors discuss these two methods in some detail. The goal of their article is to explore certain aspects of the interaction between lines, trees, and branch spaces and to obtain some results related to the problem ‘Which lines can be realized as the branch spaces of nice trees?’ Among other things, they characterize those situations in which a LOTS \(X\) embeds in the branch space of a tree constructed using convex subsets of \(X.\) Furthermore they give examples of ‘nice’ properties that a tree might have and list the topological consequences for their branch spaces. Moreover, as one typical example of the interaction between ordered spaces and trees, they characterize hereditary ultraparacompactness in a LOTS (or a generalized ordered space) \(X\) in terms of the possibility of embedding the space \(X\) in the branch space of a certain kind of tree.

Reviewer: Hans Peter Künzi (Rondebosch)

### MSC:

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |

06A05 | Total orders |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

05C05 | Trees |