Pretrees

Sets with a betweenness relation that behaves as in a tree: the axioms, intervals, the theorem that every finite pretree is a set of vertices of a finite tree, directions on linear sets, uniqueness of medians, and dense, complete and median pretrees, with an example of a complete dense pretree that is not median, separable pretrees, whose intervals are ordered like the real interval [0,1], a theorem that recognizes dendrites by a pretree structure, flows, directed arcs with their suprema, the completion of a pretree by its flows, the three kinds of flows, the adjacent points of the completion, discrete pretrees as sets of vertices of trees, quotients of pretrees, the dense quotient, the pretree of cut points of a connected space, the map from such a space to the completion of its set of cut points, and the dendrite quotient of a continuum.

Fifteen lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Betweenness and pretrees
2. Dense, complete and median pretrees
3. Separable pretrees and the order of intervals
4. Dendrites from pretrees
5. Flows on a pretree
6. Directed arcs and suprema
7. The completion of a pretree
8. Stars and the three kinds of flows
9. Adjacent points of the completion
10. Discrete pretrees and trees
11. Quotients of pretrees
12. The dense quotient of a pretree
13. Cut points of a connected space
14. Mapping a pretree to the completion of a subset
15. The dendrite quotient of a continuum

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Dendrites and cut points (Continua, arcs and cut points)
• Every path contains an arc (Continua, arcs and cut points)
• The notions of graph, path, cycle and connected graph; for the fourth lesson the course on continua, arcs and cut points and the Urysohn metrization theorem; the course proves the other facts it uses.